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>Angles>How to Determine the Angle Between Two Lines

How to Determine the Angle Between Two Lines

When you want to know the angle between two lines, you can find the angle between their directional vectors. If you have a line $l$ along the vector ${\vec{r}}_{l} = (a, b, c)$ and a line $m$ along the vector ${\vec{r}}_{m} = (d, f, g)$ , you can find the angle $α$ between the two lines this way:

Formula

The Angle Between Two Lines

\cos α = \frac{{\vec{r}}_{l} \cdot {\vec{r}}_{m}}{| {\vec{r}}_{l} | \cdot | {\vec{r}}_{m} |}, α \in [0 °, 180 °]

Note! If $α > 90 °$ , the real angle between the lines is $β = 180 ° - α$ . This is because the angle between two lines always is $\leq 90 °$ .

Example 1

You have a line

l

along the vector

{\vec{r}}_{l} = (2, 3, 4)

and a line

m

along the vector

{\vec{r}}_{m} = (1, - 2, 1)

. The angle between them is

\begin{array}{l} \cos α & = \frac{(2, 3, 4) \cdot (1, - 2, 1)}{| (2, 3, 4) | \cdot | (1, - 2, 1) |} \\ = \frac{2 \cdot 1 + 3 \cdot (- 2) + 4 \cdot 1}{\sqrt{2^{2} + 3^{2} + 4^{2}} \cdot \sqrt{1^{2} + {(- 2)}^{2} + 1^{2}}} \\ = \frac{2 - 6 + 4}{\sqrt{29} \cdot \sqrt{6}} \\ = 0, \\ α & = \cos^{- 1} (0) = \frac{π}{2} = 90 ° . \end{array}

The lines are perpendicular!

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