>Добутки та розрахунки з векторами>Для чого потрібні базисні вектори?

Для чого потрібні базисні вектори?

Знання базисних векторiв важливе для загального уявлення про вектори.

Чотири базиснi вектори та два вектори, розкладенi за допомогою базисних векторiв

Теорiя

Важливi поняття

Базис — це множина векторiв, якi в рiзних комбiнацiях утворюють решту векторiв на площинi.
Базисний вектор — це вектор iз цiєї множини.

Теорiя

Базисний вектор

Якщо дано два вектори, $\vec{x}$ i $\vec{y}$ , якi не є паралельними, то їх можна використовувати як базиснi вектори. Цi базиснi вектори можна помножити на будь-яке дiйсне число. Це означає, що решту векторiв на площинi можна записати як суму цих векторiв, помножену на число.

Приклад 1

Дано базиснi вектори $(1, - 1)$ i $(3, 4)$ . Запиши $(11, 10)$ за допомогою базисних векторiв.

\begin{array}{l} (11, 10) & = k (1, - 1) + l (3, 4) \\ = (k, - k) + (3 l, 4 l) \end{array}

(11, 10) = k (1, - 1) + l (3, 4) = (k, - k) + (3 l, 4 l)

\begin{array}{l} k + 3 l & = 11 \\ k & = 11 - 3 l \\ - k + 4 l & = 10 \\ ⇓ \\ - (11 - 3 l) + 4 l & = 10 \\ - 11 + 3 l + 4 l & = 10 \\ 7 l & = 21 | : 7 \\ l & = 3 \\ k & = 11 - 3 l \\ ⇓ \\ k & = 11 - 3 \cdot 3 \\ k & = 2 \end{array}

\begin{array}{l} k + 3 l & = 11 & - k + 4 l & = 10 \\ k & = 11 - 3 l \\ - (11 - 3 l) + 4 l & = 10 \\ - 11 + 3 l + 41 & = 10 \\ 7 l & = 21 | : 7 \\ l & = 3 \\ k & = 11 - 3 \cdot 3 \\ k & = 2 \end{array}

Тодi можемо записати

\begin{array}{l} (11, 10) & = 2 (1, - 1) + 3 (3, 4) \\ = (2, - 2) + (9, 12) \end{array}

(11, 10) = 2 (1, - 1) + 3 (3, 4) = (2, - 2) + (9, 12)

Як бачимо, вектор

(11, 10)

можна записати як суму двох базисних векторiв, помножену на константи 2 i 3.

Приклад 2

Дано вектори

\begin{array}{l} u \to = 2 \vec{a} - \vec{b}, \vec{v} & = - 6 \vec{a} - 2 \vec{b} . Знайди кут мiж векторами \vec{u} i \vec{v}, якщо | \vec{a} | = 2, | \vec{b} | = 3 i \vec{a} \cdot \vec{b} = 3Кут мiж векторами \vec{u} i \vec{v} \cos α = \frac{\vec{u} \cdot \vec{v}}{| \vec{u} | \cdot | \vec{v} |} Спочатку потрiбно знайти \vec{u} \cdot \vec{v} : \begin{array}{l} \vec{u} \cdot \vec{v} & = (2 \vec{a} - \vec{b}) \cdot (- 6 \vec{a} - 2 \vec{b}) \\ = - 12 {\vec{a}}^{2} - 4 \vec{a} \cdot \vec{b} + 6 \vec{a} \cdot \vec{b} + 2 {\vec{b}}^{2} \\ = - 12 {\vec{a}}^{2} + 2 \vec{a} \cdot \vec{b} + 2 {\vec{b}}^{2} \\ = - 12 {| \vec{a} |}^{2} + 2 \vec{a} \cdot \vec{b} + 2 {| \vec{b} |}^{2} \\ = - 12 \cdot 2^{2} + 2 \cdot 3 + 2 \cdot 3^{2} \\ = - 48 + 6 + 18 \\ = - 24 \end{array} Також потрiбно обчислити довжину \vec{u} i \vec{v} . Пам’ятай,  що {\vec{a}}^{2} = {| \vec{a} |}^{2} . \begin{array}{l} {| \vec{u} |}^{2} & = (2 \vec{a} - \vec{b}) \cdot (2 \vec{a} - \vec{b}) \\ = 4 {| \vec{a} |}^{2} - 4 \vec{a} \cdot \vec{b} + {| \vec{b} |}^{2} \\ = 4 \cdot 2^{2} - 4 \cdot 3 + 3^{2} \\ = 16 - 12 + 9 \\ = 13 \\ | \vec{u} | & = \sqrt{{| \vec{u} |}^{2}} = \sqrt{13}, \\ {| \vec{v} |}^{2} & = (- 6 \vec{a} - 2 \vec{b}) \cdot (- 6 \vec{a} - 2 \vec{b}) \\ = 36 {| \vec{a} |}^{2} - 24 \vec{a} \cdot \vec{b} + 4 {| \vec{b} |}^{2} \\ = 36 \cdot 2^{2} - 24 \cdot 3 + 4 \cdot 3^{2} \\ = 144 - 72 + 36 \\ = 108 \\ | \vec{v} | & = \sqrt{{| \vec{v} |}^{2}} = \sqrt{108} . \end{array} Тепер можемо пiдставити вирази назад у формулу, щоб знайти кут. Кут мiж \vec{u} i \vec{v} \begin{array}{l} \cos α & = \frac{- 24}{\sqrt{13} \cdot \sqrt{108}} \approx - 0.64, \\ α & \approx \cos^{- 1} (- 0.64) \approx 130.1 ° . \end{array}Попередня стаття Що таке одиничний вектор? Наступна стаття Як знайти медіани трикутника за допомогою векторівПовернутисяДодому Навчальний план Статистика ЛігаПро нас Про House of Math Співробітники Працевлаштування в House of Math Медіа Лекції Контактна інформація support@houseofmath.com Адреса House of Math 470 Ramona Street Palo Alto, CA 94301 USA FAQ Корисні посилання WeTakeAction Джуніор-матем Вправи з ручкою та папером Енциклопедія Умови Політика конфіденційності Політика щодо файлів cookies Завантажте наш додаток{"props":{"pageProps":{"campaign":false,"articleBody":" Знання базисних векторiв важливе для загального уявлення про вектори. Теорiя Важливi поняття Базис — це множина векторiв, якi в рiзних комбiнацiях утворюють решту векторiв на площинi. Базисний вектор — це вектор iз цiєї множини. Теорiя Базисний вектор Якщо дано два вектори, x→ i y→, якi не є паралельними, то їх можна використовувати як базиснi вектори. Цi базиснi вектори можна помножити на будь-яке дiйсне число. Це означає, що решту векторiв на площинi можна записати як суму цих векторiв, помножену на число. Приклад 1 Дано базиснi вектори (1,−1) i (3, 4). Запиши (11, 10) за допомогою базисних векторiв. (11, 10) = k (1,−1) + l (3, 4) = (k,−k) + (3l, 4l) (11, 10) = k (1,−1) + l (3, 4) = (k,−k) + (3l, 4l) k + 3l = 11 k = 11 − 3l − k + 4l = 10 ⇓ − (11 − 3l) + 4l = 10 − 11 + 3l + 4l = 10 7l = 21| : 7 l = 3 k = 11 − 3l ⇓ k = 11 − 3 ⋅ 3 k = 2 k + 3l = 11 − k + 4l = 10 k = 11 − 3l −(11 − 3l) + 4l = 10 − 11 + 3l + 41 = 10 7l = 21| : 7 l = 3 k = 11 − 3 ⋅ 3 k = 2 Тодi можемо записати (11, 10) = 2 (1,−1) + 3 (3, 4) = (2,−2) + (9, 12) (11, 10) = 2 (1,−1) + 3 (3, 4) = (2,−2) + (9, 12) Як бачимо, вектор (11, 10) можна записати як суму двох базисних векторiв, помножену на константи 2 i 3. Приклад 2 Дано вектори u→ = 2a→ −b→, v→ = −6a→ − 2b→. Знайди кут мiж векторами u→ i v→, якщо |a→| = 2, |b→| = 3 i a→ ⋅b→ = 3 Кут мiж векторами u→ i v→ cos ⁡ α = u→ ⋅v→ |u→| ⋅|v→| Спочатку потрiбно знайти u→ ⋅v→: u→ ⋅v→ = (2a→ −b→) ⋅(−6a→ − 2b→) = −12a→2 − 4a→ ⋅b→ + 6a→ ⋅b→ + 2b→2 = −12a→2 + 2a→ ⋅b→ + 2b→2 = −12 |a→|2 + 2a→ ⋅b→ + 2 |b→|2 = −12 ⋅ 22 + 2 ⋅ 3 + 2 ⋅ 32 = −48 + 6 + 18 = −24 Також потрiбно обчислити довжину u→ i v→. Пам’ятай, що a→2 = |a→|2. |u→|2 = (2a→ −b→) ⋅ (2a→ −b→) = 4 |a→|2 − 4a→ ⋅b→ + |b→|2 = 4 ⋅ 22 − 4 ⋅ 3 + 32 = 16 − 12 + 9 = 13 |u→| = |u→ | 2 = 13, |v→|2 = (−6a→ − 2b→) ⋅ (−6a→ − 2b→) = 36 |a→|2 − 24a→ ⋅b→ + 4 |b→|2 = 36 ⋅ 22 − 24 ⋅ 3 + 4 ⋅ 32 = 144 − 72 + 36 = 108 |v→| = |v→ | 2 = 108. Тепер можемо пiдставити вирази назад у формулу, щоб знайти кут. Кут мiж u→ i v→ cos ⁡ α = − 24 13 ⋅108 ≈−0.64, α ≈ cos ⁡ −1 (−0.64) ≈ 130.1°. ","topic":{"breadcrumbs":[{"_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","h1":"Енциклопедія","path":"/encyclopedia"},{"_key":"5ffbac10-b7fe-518f-8182-7341ca5cdebf","h1":"Числа та величини","path":"/encyclopedia/chysla-ta-velychyny"},{"_key":"5dc26fae-45dc-5c32-b6d5-036605f552ce","h1":"Вектори","path":"/encyclopedia/chysla-ta-velychyny/vektory"},{"_key":"625ba225-0a16-50c8-9478-a7625dd3eafb","h1":"Двовимірний простір","path":"/encyclopedia/chysla-ta-velychyny/vektory/dvovymirnyy-prostir"},{"_key":"69e08ae4-be71-5884-9b55-5172c4004098","h1":"Добутки та розрахунки з векторами","path":"/encyclopedia/chysla-ta-velychyny/vektory/dvovymirnyy-prostir/dobutky-ta-rozrakhunky-z-vektoramy"},{"_key":"22d10ea9-ef31-5e85-9193-d185f41950f8","h1":"Для чого потрібні базисні вектори?","path":"/encyclopedia/chysla-ta-velychyny/vektory/dvovymirnyy-prostir/dobutky-ta-rozrakhunky-z-vektoramy/dlya-choho-potribni-bazysni-vektory"}],"cssUrl":"/8c2ce5fa-9e80-4618-888c-fb8e70afc5d1-uk-5.css","h1":"Для чого потрібні базисні вектори?","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     Знання базисних векторiв важливе для загального уявлення про вектори. \u003c/p\u003e \u003cdiv class=\"center\"  \u003e \u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/vektorbasisgammel.svg\" alt=\"Чотири базиснi вектори та два вектори, розкладенi за допомогою базисних векторiв\" width=\"451\" height=\"451\"\u003e\u003c/p\u003e\u003c/div\u003e                                                                                                                                                                                                                                               \u003cdiv class=\"tcolorbox fooinfo\" id=\"tcolobox-226\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Теорiя\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"info-container-title\"\u003e \u003cspan  class=\"larm-2074\"\u003eВажливi\u003c/span\u003e \u003cspan  class=\"larm-2074\"\u003eпоняття\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e       \u003cul class=\"itemize1\"\u003e       \u003cli class=\"itemize\"\u003e       \u003cp class=\"noindent\" \u003eБазис        —        це        множина        векторiв,        якi        в        рiзних        комбiнацiях        утворюють        решту        векторiв        на        площинi.        \u003c/p\u003e\u003c/li\u003e        \u003cli class=\"itemize\"\u003e        \u003cp class=\"noindent\" \u003eБазисний        вектор        —        це        вектор        iз        цiєї        множини. \u003c/p\u003e        \u003c/li\u003e\u003c/ul\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                \u003cdiv class=\"tcolorbox fooinfo\" id=\"tcolobox-227\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Теорiя\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"info-container-title\"\u003e \u003cspan  class=\"larm-2074\"\u003eБазисний\u003c/span\u003e \u003cspan  class=\"larm-2074\"\u003eвектор\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eЯкщо дано два вектори, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e i \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e, якi не є паралельними, то їх можна використовувати як базиснi вектори. Цi базиснi вектори можна помножити на будь-яке дiйсне число. Це означає, що решту векторiв на площинi можна записати як суму цих векторiв, помножену на число. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-228\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Приклад 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003eДано базиснi вектори \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e i \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e. Запиши \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e за допомогою базисних векторiв. \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                            \u003cdiv class=\"lowerbox\"\u003e                                                                                   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class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmi  \u003ek\u003c/mi\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmi  \u003el\u003c/mi\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-label\"\u003e                                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e  \u003cp class=\"noindent\" \u003e\u003c/div\u003e \u003cdiv class=\"wide\"\u003e  \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e  \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                     \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmi  \u003ek\u003c/mi\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmi  \u003el\u003c/mi\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e  \u003cp class=\"noindent\" \u003e\u003c/div\u003e  \u003cdiv class=\"tight\"\u003e  \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                     \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mtd\u003e                                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                                          \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e                                     \u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmi  \u003ek\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                                          \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\"  type=\"implication\"\u003e⇓\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-punc\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmn\u003e7\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                                         \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                                          \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e                                     \u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mtd\u003e                                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                                          \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\"  type=\"implication\"\u003e⇓\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mtd\u003e                                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mtd\u003e                                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e  \u003cp class=\"noindent\" \u003e\u003c/div\u003e \u003cdiv class=\"wide\"\u003e  \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                 \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                         \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmi  \u003ek\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e                 \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mtd\u003e                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e                 \u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e                 \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mtd\u003e                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e                 \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-punc\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmn\u003e7\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e                 \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003el\u003c/mi\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e                 \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mtd\u003e                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e                 \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ek\u003c/mi\u003e\u003c/mtd\u003e                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                           \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e  \u003cp class=\"noindent\" \u003e\u003c/div\u003e  Тодi можемо записати  \u003cdiv class=\"tight\"\u003e  \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                    \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-label\"\u003e                                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e  \u003cp class=\"noindent\" \u003e\u003c/div\u003e \u003cdiv class=\"wide\"\u003e  \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e  \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                            \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e  \u003cp class=\"noindent\" \u003e\u003c/div\u003e  Як бачимо, вектор \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e  можна записати як суму двох базисних векторiв, помножену на константи 2 i 3.  \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e                                                                                                                                \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-229\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Приклад 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003eДано вектори \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                          \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-label\"\u003e                              \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003eЗнайди кут мiж векторами \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e i \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e, якщо \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e i \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                       \u003cp class=\"noindent\" \u003e\u003ca href=\"/uk/encyclopedia/chysla-ta-velychyny/vektory/dvovymirnyy-prostir/dobutky-ta-rozrakhunky-z-vektoramy/yak-znayty-kut-mizh-dvoma-vektoramy\"\u003eКут\u003c/a\u003e мiж векторами \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e  i \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e  \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e  \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                   \u003cmi class=\"qopname\"\u003ecos\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eα\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e  \u003cmfrac\u003e\u003cmrow  \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e  \u003cp class=\"noindent\" \u003eСпочатку потрiбно знайти \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math type=\"divide\"\u003e:  \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                               \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                          \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                          \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e  \u003cp class=\"noindent\" \u003eТакож потрiбно обчислити довжину \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e  i \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e. Пам’ятай,  що \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/math type=\"punct\"\u003e.  \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                  \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                          \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                         \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                          \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e \u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e  \u003cp class=\"noindent\" \u003eТепер можемо пiдставити вирази назад у формулу, щоб знайти кут. Кут мiж  \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e i  \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e  \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                  \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e cos\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  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Вивчай математику швидко та легко. 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Повертайся сюди, щоб переглянути тести та оновлення своєї математичної подорожі. Там ти знайдеш такі речі:","chestTitle":"Ласкаво просимо!","close":"Закрити","collection":"Моя колекція","collectionList":{"allItems":"Усі кубки","allRewards":"Усі нагороди","bootcamps":"Bootcamps","competition":"Змагання","mathdrill":"Основи математики","mathEssentials":"Основи математики"},"collectionShowcase":{"bootcamp":"Bootcamp","gathered":"Зібрано","championship-2021":"Чемпіонат з множення 2021"},"rankPopupTitle":"Нова система підвищення рівня XP!","rankPopupText":"Ти можеш здобути XP, виконуючи завдання на порталі. Досягнувши порогу XP, ти підвищиш рівень.","completed":"Завершено","weExchange":"Обмінюємо","dummyData":{"mult":{"title":"Множення","description":"Що таке множення?"},"algebra":{"title":"Алгебра","description":"Елементарна алгебра"}},"event":{"checkQuest":"Прочитано та зрозуміло","INTRO":{"dialogue":[{"text":"Це твоя база. Тут ти отримуватимеш повідомлення та завдання, збираючи скарби та нагороди під час своєї подорожі світом математики.","title":"Ласкаво просимо додому"}]}},"findYourLearningPath":"Знайди свій шлях навчання, створи аватар та отримай домашнього дракона!","gemBlueAlt":"Блискучий синьо-білий діамант","gemGreenAlt":"Блискучий зелений смарагд","gemRedAlt":"Сяйнистий червоний рубін","gotoavatarpage":"Перейти до аватара","Home":"Домашня база","homebasepage":"Початкова база:","imageAlt":{"goldenHex":"Золотий шестикутник","levelBadge":"Значок переходу на вищий рівень"},"introText":"Ти саме той герой/героїня, якого ми шукаємо. Ми повернемося до цього. Дозволь спочатку показати тобі все навколо.","introTitle":"Гарний початок!","kudos":"Kudos","level":"РІВЕНЬ","levelUp":"Підвищення рівня","navbarIcon":"Схованка","next":"Далі","shadow":"Тінь","playerName":"Ім'я гравця...","questCompleted":" Тест виконано!","questDescriptions":{"bootcamp":" Досліди bootcamp \u003cb\u003e{{BootcampTitle}}\u003c/b\u003e і здобудь перемогу над босом буткемпів!","essential":"Щоб підтвердити свої навички, пройди Основи математики!","encyclopedia":"Прочитай і зрозумій \u003cb\u003e{{EncyclopediaArticle}}\u003c/b\u003e  та постав відповідну позначку!"},"quests":"Мій математичний тест","mathQuests":"Математичні тести","completedMathQuests":"Виконані завдання","completeQuestsToSee":"Щоб побачити тести тут, потрібно їх виконати","questText":"Пройди діагностичний тест й отримай індивідальний шлях навчання","quiz":{"advancedQuizText":"Щоб рухатися вперед, тепер тобі потрібно лише обрати теми, які тобі знайомі.","advancedQuizTitle":"Чудові результати тесту!","advancedSubjects":{"quiz-subject-advanced-derivatives":{"title":"Похідна"},"quiz-subject-advanced-differential-equations":{"title":"Диференціальні рівняння"},"quiz-subject-advanced-integrals":{"title":"Інтеграли"},"quiz-subject-advanced-logarithms":{"title":"Логарифми"},"quiz-subject-advanced-pi":{"title":"Число «Пі», π"},"quiz-subject-advanced-probability":{"title":"Теорій ймовірностей"},"quiz-subject-advanced-proofs":{"title":"Доведення"},"quiz-subject-advanced-quadratic-equations":{"title":"Квадратні рівняння"},"quiz-subject-advanced-series":{"title":"Числові ряди"},"quiz-subject-advanced-sets-of-numbers":{"title":"Числові множини"},"quiz-subject-advanced-trigonometry":{"title":"Тригонометрія"},"quiz-subject-advanced-vectors":{"title":"Вектори"}},"applyText":"Ти, очевидно, добре знаєш математику! Якщо ти бажаєш доєднатися до команди House of Math, будь ласка, подай заявку за посиланням \u003clinkToCareers\u003ehere\u003c/linkToCareers\u003e.","awardTitle":"Ми перетворюємо твій досвід навчання на гру","awardTitleSubtext":"Під час мандрівки ти знайдеш такі скрині з нагородами.","awesome":"Чудово!","buttonTakeQuiz":"Пройти тест","creatingYourLearningPath":"Створення твого шляху навчання...","learningPath":"Створення твого шляху навчання","greatResultsText":"Щоб рухатися вперед, тепер тобі потрібно лише обрати теми, які тобі знайомі.","greatResultsTitle":"Чудові результати тесту!","subjectKnowledgeRating":{"0":"Жахливо","1":"Погано","2":"Нейтрально","3":"Добре","4":"Чудово"},"subjects":{"quiz-subject-algebraic-factorization":{"questionRating":"Наскільки добре ти опанував/опанувала розкладання многочлена на множники?","title":"Розкладання многочлена на множники"},"quiz-subject-basic-algebra":{"questionRating":"Наскільки добре ти опанував/опанувала операції з математичними знаками?","title":"Основи алгебри"},"quiz-subject-equations":{"questionRating":"Наскільки добре ти опанував/опанувала рівняння?","title":"Рівняння"},"quiz-subject-fractions":{"questionRating":"Наскільки добре ти опанував/опанувала дроби?","title":"Дроби"},"quiz-subject-functions":{"questionRating":"Наскільки добре ти опанував/опанувала функції?","title":"Функції"},"quiz-subject-parentheses":{"questionRating":"Наскільки добре ти опанував/опанувала круглі дужки?","title":"Круглі дужки"},"quiz-subject-percent":{"questionRating":"Наскільки добре ти опанував/опанувала відсоток?","title":"Відсоток"},"quiz-subject-powers":{"questionRating":"Наскільки добре ти опанував/опанувала степінь числа?","title":"Степінь числа"},"quiz-subject-the-multiplication-table":{"questionRating":"Наскільки добре ти опанував/опанувала таблицю множення?","title":"Таблиця множення"}},"title":"Щоб навчання було ефективним, пройди тест!","titleSubtext":"Пройди цей тест, щоб дізнатися, з чого почати.","yesNoQuestionTitle":"Чи працював/працювала ти коли-небудь над \u003clowercase\u003e{{ subject }}\u003c/lowercase\u003e?"},"retakeQuiz":"Повторити тест","reward":"Нагорода","storyCards":{"buttonCool":"Класно!","buttonIWill":"Я буду!","buttonJump":"Стрибай","buttonLetUsTry":"Спробуймо!","buttonNice":"Чудово!","buttonOfCourse":"Звісно!","buttonOh":"О!","buttonPeekaBoo":"Піка Бу","buttonRoar":"Рев","buttonThanks":"Дякую!","buttonWow":"Овва!","buttonYes":"Так!","ChoosePetTrick":"Твій дракон готовий навчитися нового трюку! Обери один із трьох нижче!","ChooseYourEgg":"Дивися, хтось залишив тобі подарунок!","CoolDown":"Тобі потрібно зачекати певний час, перш ніж ти знову зможеш віддати математичну енергію! Використай цей час, щоб зібрати більше енергетичних сфер в  Основах математики","CoolDownFirstTime":"Твоє яйце повне, й потрібно трохи зачекати, перш ніж можна буде віддати більше математичної енергії! Використай цей час, щоб зібрати більше математичної енергії зі скринь в Основах математики!»","DoNotHaveEnoughMnO":"Ти більше не маєш енергії лайків! Щоб здобути більше, виконай тести!","DragonFull":"Твій дракон ситий! Зачекай певний час, перш ніж давати йому енергію лайків.","DragonTrickCooldown":"Твій дракон утомився після навчання нового трюку. Перш ніж пробувати знову, дозволь йому трохи відпочити!","DragonTrickNotWilling":"Твій дракон не бажає тренуватися, доки не буде щасливий і нагодований!","DragonTutorial1":"Дбай про здоров’я свого дракона, годуючи його енергією лайків!","DragonTutorial2":"Ходять чутки, що якщо ти деякий час підтримаєш свого Дракона здоровим, він може навчитися нових трюків!","DragonTutorial3":"Дракони також потребують пестощів! Регулярно гладь свого дракона!!","DragonTutorial4":"Молодець! Твій Дракон щасливий! Продовжуй дарувати йому любов щодня!","EggHasBeenChosen":"Чудовий вибір! Пропоную безпечно помістимо яйце у твою базу. Чи зможемо ми змусити його вилупитися?","EggHungry1":"Гм... . Твоє яйце здається голодним!","EggHungry2":"Ось, візьми цю енергію лайків!","EggHungry3":"Спробуй нагодувати яйце енергією лайків, натиснувши на нього!","EggInfoStillHatching":"Твоє яйце зараз вилуплюється! Проте потрібно згодувати йому ще енергії лайків!","EggIsAboutToHatch":"Неймовірно! Тепер ти маєш Дракона на своїй домашній базі","FeedMathEnergyToEggFirstTime":"Спробуй нагодувати яйце енергією лайків, натиснувши на нього!","FirstTimeGettingMathEnergyFromChest":"Круто! Ти знайшов свою першу енергетичну сферу! Віддай її своєму яйцю на домашній базі!!","GiveDragonName":"Введи ім'я дракона або просто натисни OK.","InfoAboutPetTrick":"Твій дракон може вчитися, коли він ситий і задоволений!","MathEnergyTutorial1":"Щоб яйце вилупилося, потрібно нагодувати його достатньою кількістю енергії лайків.","MathEnergyTutorial2":"Щоб здобути більше енергії лайків, виконай тести!","NextHatchingState1":"Овва! Глянь! Твоє яйце тепер має інший вигляд!","NextHatchingState2":"Твоє яйце тріснуло ще трохи! Цікаво, що всередині...","NextHatchingState3":"Ще трохи... Твоє яйце скоро вилупиться!","TimesTableCompetition2021FirstPlace":"Вітаємо з перемогою в Чемпіонаті з множення 2021 року! Отримай унікальний кубок і додай його до своєї колекції.","TimesTableCompetition2021Participant":"Дивовижна робота! Ти заробив/заробила цей приз, беручи участь у Чемпіонаті з множення 2021 року.","TimesTableCompetition2021SecondPlace":"Вітаємо з 2-м місцем у Чемпіонаті з множення 2021 року! Отримай унікальний кубок і додай його до своєї колекції.","TimesTableCompetition2021ThirdPlace":"Вітаємо з 3-м місцем у Чемпіонаті з множення 2021 року! Отримай унікальний кубок і додай його до своєї колекції."},"space":"Пробіл","takeTheQuiz":"Пройди тест, щоб розблокувати свій дім!","tooltip":{"videoCourse":"Bootcamp","encylopedia":"Енциклопедія","mathEssential":"математичні вишколи","description":{"videoCourse":"Дивися відео, виконуй завдання та здобудь перемогу над босом буткемпів","encylopedia":"Прочитай і зрозумій та постав відповідну позначку!","mathEssential":"Щоб підтвердити свої навички, пройди Основи математики!"}},"titles":{"houseOfMathMember":"Член House of Math","mathApprentice":"Математик-практикант","mathemagician":"Матемаг","mathJunior":"Математик-початківець","mathlete":"Мататлет","mathScholar":"Математик-експерт","mathSorcerer":"Математик-чудотворець"},"unlockYourHomebase":"Увійди або зареєструйся, щоб перейти до особистого кабінету, де можна створити свій індивідуальний шлях навчання, відстежувати здобуті бали досвіду (XP) та винагороди, отримати доступ до своєї Схованки й багато іншого!","updateQuests":"Оновити навчальні тести","xp":"XP","open":"ВІДЧИНЕНО","yourCompletedQuests":"Твої виконані квести","yourFirstQuest":"Твій перший квест","yourNextQuest":"Твій наступний квест","yourUpcomingQuests":"Твої майбутні квести","removeTask":"Видалити зі шляху навчання","cancel":"Скасувати","rank":"Ранг","leagueNavButton":"Ліга","menuLabel":"Меню","loading":"Завантаження","loadscreenHint":{"feedDragon":"Підвищ рейтинг свого дракона, годуючи його енергією лайків","earnKudos":"Заробляй енергію лайків, виконуючи математичні завдання","earnGems":"Заробляй коштовності, виконуючи математичні завдання","buyStuff":"Купуй круті речі з коштовностями в крамниці дракона","wardrobe":"Усі куплені речі зберігаються в шафі","moreMath":"Чим більше математичних завдань ти виконаєш, тим більше отримаєш коштовностей та енергії лайків"}},"menu":{"mathTopics":"Уся математика","bootcamp":"Bootcamps","dashboard":"Додому","digitalTools":"Цифрові засоби","drillhefter":"Основи математики","encyclopedia":"Енциклопедія","exams":"Eksamensoppgaver","exitPreview":"Закрити попередній перегляд","games":"Ігри","logOut":"Вийти","login":"Увійти","mathEssentials":"Основи математики","mathVault":"Джуніор-матем","register":"Зареєструватися","settings":"Налаштування","studyTips":"Поради з навчання","toggleNavigationMenu":"Перемкнути меню навігації","tutors":"Tutor-on-Demand","daysStreak":"{{days}} днів з реєстрації","learningTools":"Навчальні матеріали","moreLearningTools":"Більше навчальних матеріалів","penNPaperExercise":"Вправи з ручкою та папером","excelOrGeoGebra":"Рецепти Excel / GeoGebra","numberNQuantities":"Числа та величини","statNProbability":"Статистика та ймовірність","stackNLoad":"Stack n´load","juniorMath":"Джуніор-матем","windSurfer":"Wind Surfer","treasureTrail":"Treasure Trail","multiplicationMaster":"Multiplication Master","misc":"Профіль","showMore":"Показати більше","showAllGames":"Показати всі ігри","gotoGames":"Перейти до ігор","switchProfile":"Перемкнути профіль","createFamily":"Додати профіль","getHomApp":"Завантажте додаток House of Math","tryFree":"Спробуйте безкоштовно","signUpForFree":"Зареєструватися безплатно","referral":"Запросити друзів"},"navbar":{"explore":"Ознайомитися","menu":"Меню","panelAdmin":"Панель адміністратора","panelTutor":"Панель репетитора","panelStudent":"Учнівська панель","search":"Пошук","languages":"мова","norwegian":"Norsk","english":"English","ukrainian":"Українська","settings":"Налаштування","logOut":"Вийти","dashboard":"Додому","chat":"Чат","userProfile":"Профіль","leaderboard":"Список лідерів"},"notFound":{"clickOne":"Клацни людину, яку, на твою думку, варто звільнити","fire":"Звільнити","goDashboard":"Перейти до панелі керування","pageNotFound":"Сторінку не знайдено","pageRemoved":"Сторінку видалено","punishDevs":"Наших розробників потрібно покарати за це жахливе недбальство","employeesOnLunchBreak":"Здається, наші співробітники пішли на обідню перерву. Ми незабаром повернемося!"},"search":{"adaptedView":"Індивідуальний режим перегляду","bootcamp":"Bootcamps","all":"Усе","drill":"Основи математики","encyclopedia":"Енциклопедія","noResult":"Результатів пошуку не знайдено","readMore":"Дізнатися більше","watchVideo":"Дивитися відео","remove":"Видалити","search":"Пошук..","geogebra":"GeoGebra","searchResultAppear":"Тут з’являться ваші результати пошуку","searchTooltip":"Індивідуальний режим перегляду призначений для перегляду статей рівня, який ти обрав/обрала у налаштуваннях. Перемкни, щоб переглядати всі статті.","software":"Цифрові засоби","title":"Результати пошуку","searchPlaceholderText":"Чого тобі хочеться вивчити?"},"dashboard":{"activity":"Набір завдань","bootcamp":{"buttonContinueLearning":"Продовжити навчання","buttonToOverview":"Перейти до огляду","comingSoon":"Незабаром!","edit":"Редагувати","title":"Мої Bootcamps"},"gamification":{"badge":"Значок Bootcamp","badgeBackground":"Фон бейджа","buttonToBase":"Схованка","gemIcon":"Коштовності Значок","completeQuestsGetTopics":"Виконай тести, щоб отримати доступ до більшої кількості тем"},"completeProfile":{"button":"Завершити реєстрацію","description":"Нам потрібно дізнатися про тебе трохи більше, щоб персоналізувати твій досвід"},"continue":"Продовжити","topicOverview":"Огляд теми","tutor":{"assigningTutor":"Будь ласка, зачекай, доки ми знайдемо для тебе репетитора.","bookPrivateTutoring":"Замовити репетиторство","tutorPending":"Очікуй","tutorScheduling":"Коли тобі буде призначене заняття, воно з’явиться тут.","myTutor":"Мій репетитор","myTutors":"Мої репетитори","myStudentPanel":"Моя учнівська панель"},"activities":"Завдання","myActivities":"Мої завдання","tagNew":"новий","news":{"buttonChat":"Чат","buttonPlay":"Грати","buttonShowMe":"Покажи мені","buttonStart":"Розпочати","campAlt":"Зона Bootcamps","content1":"Долучайся до пригод разом із бджілкою Д’Рілло за допомогою оновленого розділу Основи математики. Дізнайся про найважливіші математичні теми, граючи з веселими завданнями!","content2":"Наші репетитори готові тобі допомогти. Скористайся чатом, щоб отримати відповіді на всі свої запитання з математики.","content3":"Візьми участь у Чемпіонаті з множення, змагайся з іншими та вигравай першокласні призи!","content4":"Bootcamps — це веселі, інтерактивні, анімовані уроки з математики в ігровій формі для всіх.","content5":"Отримай миттєву допомогу з математики! Замов 15-хвилинне відеозаняття з репетитором.","header1":"Стань майстром математичного вишколу","header2":"Зруш з місця","header3":"Виграй iPad та інші чудові призи ","header4":"Спробуй Bootcamps","header5":"Відеочат ","subject1":"Дроби","title":"Новини"},"newsInHomebase":"Новини на домашній базі","metaDescription":"Уся необхідна тобі допомога з математики безплатно! Отримай повний навчальний план із математики разом із навчальними відео, інтерактивним розв’язуванням завдань, іграми та цілодобовим чатом.","takeQuiz":"Пройти тест","tipOftheDay":"ПОРАДА ДНЯ","doMath":"Зайнятися математикою","doMathLT":"Займайся математикою!","learningPath":"Шлях навчання","learningPathTutorial":{"step1":{"title":"Ми створили твій \u003cspan\u003eшлях навчання\u003c/span\u003e на підставі твоїх відповідей.","action":"Далі"},"step2":{"title":"Щоб пройти рівень, переглянь \u003cspan\u003eвідео\u003c/span\u003e та виконай \u003cspan\u003eзавдання\u003c/span\u003e","action":"Дізнатися більше"},"step3":{"title":"Оціни свої знання,\u003cbr /\u003e взявши участь у \u003cspan\u003eбитві з босом\u003c/span\u003e","action":"Далі!"},"step4":{"title":"Ви завжди можете отримати нові завдання, оновивши ваш \u003cspan\u003eшлях навчання\u003c/span\u003e.","action":"Далі"},"step5":{"title":"Щоб ознайомитися з усіма нашими матеріалами, відвідай розділ \u003cspan\u003eУся математика\u003c/span\u003e","action":"Закінчити"}},"curriculumTutorial1":{"step1":{"title":"Кожна академічна шапочка \u003cCapImage /\u003e позначає математичну тему. 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Students in Texas, Virginia, Alaska, and Nebraska can also achieve strong learning outcomes with House of Math, but should use the our AI Oracle to ensure they cover all required topics. The platform supports all K–12 math globally, as well as selected university-level subjects.","cancelAnytime":"Yes. You can cancel whenever you want, and you'll keep access until the end of your current billing period. No hidden fees.","gradeLevelsCover":"We cover the full K–12 U.S. curriculum. Lessons are reviewed and verified by expert educators.","freeTrial":"Yes, we offer a 7 day free trial. Cancel anytime during the trial and you won't be charged."}}}},"getStartedV2":{"freeTakes90Seconds":"Free · Takes 90 seconds","understandChildMath":"Let's understand what's happening with your child’s math","answerQuestions":"Answer a few quick questions about what you're seeing at home. 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The real cause is usually a few missing foundational blocks that quietly make every lesson feel scary. House of Math finds exactly which blocks need rebuilding.","ES_EMO_NEW":"Seeing your child shrink away from math is heartbreaking, especially this young. When {{breakdown}} starts to feel scary, it often traces back to a few foundational blocks that never fully solidified — not to lack of effort. House of Math finds exactly which blocks are missing and rebuilds them.","ES_ACA_EXP":"You've already invested in outside support — yet progress in {{breakdown}} hasn't matched the effort. More hours rarely fixes this; finding the specific foundational blocks your child skipped does. House of Math identifies exactly those gaps and rebuilds them.","ES_ACA_NEW":"Early math foundations shape everything that comes later, so wanting your child to do well now is exactly the right instinct. When {{breakdown}} feels harder than it should, the cause is almost never effort — it's a few foundational blocks that didn't fully land. House of Math finds and rebuilds exactly those blocks.","ES_DIA_EXP":"You can tell something isn't working — even after trying outside help — but pinpointing what is harder than people admit. At this age, vague struggles almost always trace back to a few specific foundational blocks, often around {{breakdown}}. House of Math diagnoses exactly which blocks need attention and rebuilds them.","ES_DIA_NEW":"Sensing something is off — without knowing exactly what — is actually the most useful starting point. At the elementary level, vague struggles almost always come down to a few specific foundational blocks, often around {{breakdown}}. House of Math diagnoses exactly which blocks need rebuilding.","MS_EMO_EXP":"Middle school is when math anxiety hits hardest, and the extra support you've given hasn't fully reached the stress around {{breakdown}}. The pressure usually isn't about middle school content itself — it builds on earlier foundational blocks that never fully solidified. House of Math finds and rebuilds exactly those blocks, so confidence can return.","MS_EMO_NEW":"Math pressure peaks in middle school, and seeing your child shut down around {{breakdown}} is genuinely hard to watch. This anxiety almost never starts in middle school — it builds on earlier foundational blocks that quietly never solidified. House of Math finds those exact missing blocks and rebuilds them, clearing the anxiety at its source.","MS_ACA_EXP":"You've invested real time and outside support in your child's math — yet {{breakdown}} still feels harder than it should. That's not a sign of insufficient effort; it's a signal that earlier foundational blocks have gaps. House of Math finds exactly which blocks are missing and rebuilds them, so the hours invested finally translate into progress.","MS_ACA_NEW":"Middle school is a critical window — what gets solid now shapes high school and beyond. If {{breakdown}} is harder than it should be, the cause is usually a few earlier foundational blocks that didn't fully form, not effort. House of Math finds exactly those blocks and rebuilds them before middle school content can fully click.","MS_DIA_EXP":"You've tried outside help, yet something still feels off — even if you can't name exactly what. Middle school struggles almost always trace back to specific foundational blocks from earlier years, often surfacing around {{breakdown}}. House of Math diagnoses exactly which blocks are missing and rebuilds them.","MS_DIA_NEW":"Knowing something is off — without being able to pinpoint what — is actually a clear-eyed read. Middle school difficulties almost always trace back to specific foundational blocks from earlier years, often around {{breakdown}}. House of Math diagnoses exactly which blocks need rebuilding before they compound further.","HS_EMO_EXP":"High school math anxiety is uniquely heavy — grades feel permanent, and the extra support you've provided hasn't eased the stress around {{breakdown}}. High school content sits on years of earlier foundational blocks, and gaps from before are usually what's driving the anxiety now. House of Math finds those exact missing blocks and rebuilds them.","HS_EMO_NEW":"By high school, math anxiety carries real weight — grades and college pressure feed it, especially around {{breakdown}}. Pushing harder rarely helps now; the anxiety usually traces back to foundational blocks from earlier years that quietly never solidified. House of Math finds those exact blocks and rebuilds them, clearing the source of the stress.","HS_ACA_EXP":"You've invested seriously in outside support — yet {{breakdown}} still feels harder than the effort warrants. With high school stakes this high, that gap is worth taking seriously: it almost always traces to earlier foundational blocks that never fully solidified. House of Math finds those exact blocks and rebuilds them, so your child's effort finally compounds.","HS_ACA_NEW":"High school math determines real outcomes — GPA, college options. If {{breakdown}} is harder than it should be, the cause is rarely effort; high school content stacks on years of earlier foundational blocks, and gaps from before make advanced topics feel impossible. House of Math finds and rebuilds those missing blocks before the gap widens.","HS_DIA_EXP":"Knowing something is off — despite the outside help you've sought — is a more accurate read than most parents credit themselves for. 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Handy when multiple family members share the same device.","changePin":"Change PIN","connected":"\u003cbold\u003eConnected\u003c/bold\u003e {{connectedEmail}}","addPin":"Add PIN","setPin":"Set PIN","editPin":"Edit PIN","expired":"Expired","pending":"Invitation pending","days":"{{days}} days left","hours":"{{hours}} hours left","min":"{{min}} min left","cancelInvite":"Cancel invitation","resendInvite":"Resend invitation link","viewStats":"View Stats","viewProfile":"View Profile","settings":"Settings","viewing":"Viewing","userStats":"{{userName}}'s Stats","familyJoinedSuccess":"You have successfully joined the family","inviteSent":"Invitation sent","pinRemoved":"PIN removed","loggedInWithAdmin":"You are currently logged in with this family's admin account","loggedInWithEmail":"You are currently logged in with {{email}}","useTheCorrectUser":"Log out, then log in with the correct user or create a new account.","useAdminLogin":"Use admin account login","useAdminLoginDescription":"The profile will log in using the same account as the admin","sendPersonalLoginInvite":"Send invite for personal login","personalLoginInviteDescription":"The profile owner will create and set up their own login details","sendFamilyInvite":"Invite another user to join your family account","loginOptions":"How should this profile log in to House of Math?","backToAdminPanel":"Back to Admin Panel","inviteSentTo":"Invitation sent to","inviteInstructions":"The invited user must accept the invitation on their device to be included in the family account.","inviteLoginInstructions":"Once accepted, they can log in to House of Math using their preferred login method.","nextStep":"Next step:","invitedPersonNeedsTo":"The invited person needs to:","openInvitationEmail":"Open the invitation email we just sent them.","clickAcceptInvitation":"Click Accept Invitation in the email.","afterTheyAccept":"After they accept, they can log in to House of Math with their preferred login method.","administerSubscription":"Administer subscription","manageSubscription":"Manage subscription","managePayment":"Manage Payment","changeSubPlan":"Change subscription plan","addLicense":"Add more learners","removeLicense":"Remove learners","ManageFamily":"Manage subscription plan","nextBillingPeriod":"Your next billing period is on {{date}}","licensesNextPeriod":"Math access on next period","reactivate":"Who do you want to reactivate?","error":{"pinRequired":"PIN required","noPin":"No PIN has been set for this profile","incorrectPin":"Incorrect PIN","pinNotDeleted":"Failed to delete PIN","inviteCanceled":"This invitation has been canceled","inviteNotFound":"Invitation not found","inviteUsed":"This invitation has already been used","inviteExpired":"This invitation has expired","cantInviteSelf":"You cannot invite yourself","inviteAlreadySent":"Invitation already sent","inviteCodeRequired":"Invitation code required","errorCreatingInvite":"Error creating invitation","familyNotFound":"Family not found","adminNotFound":"Admin not found","userNotFound":"User not found","notSameFamily":"You are not in the same family","errorAddingToFamily":"Error adding to family","licenseLimitReached":"Math access limit reached","incorrectPassword":"Incorrect password","noPassword":"No password has been set for this profile","errorNickname":"Error updating nickname","needNickname":"Please enter a nickname","noLearnerData":"No learner data found","notPartOfFamily":"User is not part of a family account","notAdmin":"User does not have admin rights","notAddSelf":"You can't add yourself to your own family account","alreadySubscribed":"User already has a subscription","alreadyPartOfFamily":"User is already part of a family account","inviteError":"Failed to send invitation.","failedInvite":"Failed to create invitation","validationFailed":"Validation failed","pinLength":"PIN must be 4 characters long","somethingWentWrongPleaseReload":"Something went wrong, please reload.","genericError":"An error occurred. Please try again later","notFamilySubscription":"Last subscription is not a family plan","alreadyInThisFamily":"You are already in this family","invalidEmail":"Invalid email address","emailAlreadySet":"Add a family password first, then you can change the email.","emailUnchanged":"This is already your email.","adminAlreadyHasLicense":"Admin already has math access","userAlreadyHasLicense":"User already has math access","noActiveSubscription":"You don’t have an active subscription","subscriptionNotActiveOrTrialing":"To upgrade, your subscription must be active or trialing. You can either reactivate it or wait until your current cancellation period ends before upgrading.","vippsNotSupportedForFamilyUpgrade":"Family upgrade isn’t supported with Vipps payment yet. We are working on it.","alreadyOnFamilyPlan":"You’re already on a family plan.","learnerCannotUpgrade":"You’re part of a family plan and can’t upgrade. The family admin has to remove you from the family before you can upgrade.","paymentFailed":"Your payment failed. Your card details may be missing or expired.","subscriptionScheduledForCancellation":"Can not perform the action as the subscription is scheduled for cancellation.","passwordTooShort":"Password must be at least 8 characters","noEmailSet":"Add an email before setting a family password","emailInUser":"This email is already used by another account","passwordMismatch":"The passwords do not match","currentPasswordRequired":"Enter the current password"},"logoutMessage":"This user need to login with their own username and password","removingLicense":"You are removing a math access.","removedAfterPeriod":"This math access will remain active until the end of your current billing period: {{date}}","willLoseAccess":"\u003citalic\u003e{{username}}\u003c/italic\u003e associated with this math access will lose access on that date.","willNotBeDeleted":"\u003citalic\u003e{{username}}\u003c/italic\u003e will not be deleted.","pendingRemoval":"Pending removal {{date}}","removingTheLast":"You are about to remove the last member of your family group.","toProceed":"To proceed, go to the subscription management page","toManageSubscription":"Go to mange subscription page","buyLicense":"Buy math access","assignMathAccess":"Assign Math access","userAccessSuccess":"{{user}} is now a math learner","createPin":"Create admin pin","pinProtection":"Protect the admin panel by\u003cbr /\u003e setting a PIN.","setPinDescription":"Use this pin to ensure no one else can access the admin panel","connectAccount":"Do you want to connect this profile to an existing House of Math account?","ConfirmPay":"Confirm and pay","toPayNow":"To pay now","yourPlan":"Your plan","acceptAs":"You will be accepting the invite as","useOther":"Want to use another account?","subscriptionWarning":"By accepting this invitation, your subscription will be transferred to the Family Account. Please note that any remaining credit on your current active subscription will not be refunded.","confirmSubscriptionCancellation":"Accepting the invitation will cancel you current subscription, please confirm to proceed","upgradeModalWarning":"Admin access is required keep current accounts math access \u0026 other features.\n Continue with admin access?","upgrade":{"title":"Upgrade","nextBillingPeriodStarts":"Next billing period starts","currentPlan":"Current plan: ({{plan}})","newPlan":"New plan: ({{plan}})","mathLicenses":"Math accesses on next period:","yourNextBilling":"Your next billing","adjustedCredit":"Adjusted cost for the upgrade","amountToPayToday":"Amount to pay today","prorationMsg":"Credit from remaining billing period","upgradeTooltip":"Your credit is based on the unused portion of your previous payment, and the new price of the upgraded plan for the remaining time in the billing period.","upgradeDescription":"We’ve applied a prorated charge for the upgrade. Your new plan will be billed in full on the next billing date."},"yes":"Yes","logOut":"Log out","createFamilyPlan":"Create your family plan","upgradeFamilyPlan":"Upgrade to family","upgradeUserAccount":"Upgrade {{user}} to Math Learner","userUpgradeMsg":"You have {{count}} Math Accesses available. 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Would you like to buy one now?","fullOverview":"Full overview in admin panel","wantMathAccess":"Do you want math access too?","yesMathAccessMonth":"Yes, include math access for me\u003cbr/\u003e\u003cstrong\u003e(+{{price}} per month)\u003c/strong\u003e","yesMathAccessYear":"Yes, include math access for me\u003cbr/\u003e\u003cstrong\u003e(+{{price}} per year)\u003c/strong\u003e","noMathAccess":"No, parent panel only (Free)","priceBreakdown":"Pricing breakdown","familyPlanTitle":{"year":"My yearly family plan","month":"My monthly family plan"},"individualPlanTitle":{"year":"My yearly plan","month":"My monthly plan"},"trialPeriod":"{{days}}-day Free Trial","yourSub":"Your subscription plan","month":"Monthly family","year":"Yearly family","adminAccess":"Admin math access","membersAccess":"Additional learners ({{count}})","pricingCardCreditMsg":"Remaining credit after payment","basePrice":"First learner","baseAccess":"Base price","afterTrialPeriod":"After {{days}}-Day Free trial","startTrialPeriod":"Start {{days}}-Day Free trial","total":"Total","free":"Free","then":"Then","thenMonths":"For {{months}} month(s)","thenYears":"For {{years}} year(s)","currentAccountChange":"Current account will be changed into to a family account immediately after upgrade.","dataTransfer":"All data from current account will be transferred to family account","amountToPay":"Amount to pay today","upgradeAmountPay":"Amount to pay","newMonthlyTotal":"New Monthly total","howIsThisCalculated":"How is this calculated?","newYearlyTotal":"New Yearly total","subtotal":"Subtotal","totalFamilyAccountPrice":"Your total family account price","familyAccountSummary":"Family Account summary","toPayToday":"To pay today","couponValidFor":"Coupon code valid for","couponForever":"forever","couponOnce":"once","couponValidForMonths":"{{months}} month(s)","couponValidForYears":"{{years}} year(s)","couponValidForYearsAndMonths":"{{years}} year(s) and {{months}} month(s)","setUpMathAccess":"Set up all available math accesses. 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