Angle Between Lines

Topics: Geometry - Line - Angle

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Given two lines in ${\mathbb{R}}^3$, $l_1=\{P+su:s\in \mathbb{R}\}$ and $l_2=\{Q+tv:t\in \mathbb{R}\}$, we can determine the angle between them with:

$$\cos \theta =\frac{|u\cdot v|}{\Vert u\Vert \Vert v\Vert }$$

The angle $\theta$ between any two lines will always be such that $0\le \theta \le \frac{\pi }{2}$.

Comparing formulas

Notice this formula is almost the same as the formula for the Angle Between Vectors, the only difference being the fact that we take the absolute value of $u\cdot v$.

This is because two vectors of equal magnitude but opposite directions generate the same exact line (given an adequate starting point).

Orthogonality §

These two lines $l_1$ and $l_2$ are orthogonal if and only if their direction vectors $u$ and $v$ are orthogonal.

That is, two lines are orthogonal if and only if their angle $\theta =\frac{\pi }{2}$ (in other words, if $\cos \theta =0$).

Parallelism §

These two lines $l_1$ and $l_2$ are parallel if and only if their direction vectors are parallel.

That is, two lines are parallel if and only if their angle $\theta =0$ (in other words, if $\cos \theta =1)$.