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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