# Angle Between Vectors in Terms of Dot Product

## Theorem

The angle between two non-zero vector quantities can be calculated by:

$\theta = \arccos \frac {\mathbf v \cdot \mathbf w} {\norm {\mathbf v} \norm {\mathbf w} }$

where:

$\mathbf v \cdot \mathbf w$ represents the dot product of $\mathbf v$ and $\mathbf w$
$\norm {\, \cdot \,}$ represents vector length
$\arccos$ represents arccosine.

## Proof

 $\ds \norm {\mathbf v} \norm {\mathbf w} \cos \theta$ $=$ $\ds \mathbf v \cdot \mathbf w$ Cosine Formula for Dot Product $\ds \cos \theta$ $=$ $\ds \frac {\mathbf v \cdot \mathbf w} {\norm {\mathbf v} \norm {\mathbf w} }$ because $\mathbf v, \mathbf w \ne \mathbf 0 \implies \norm {\mathbf v}, \norm {\mathbf w} \ne 0$ $\ds \map \arccos {\cos \theta}$ $=$ $\ds \map \arccos {\frac {\mathbf v \cdot \mathbf w} {\norm {\mathbf v} \norm {\mathbf w} } }$ because $0 \le \theta \le \pi$ $\implies -1 \le \cos \theta \le 1$ $\ds \theta$ $=$ $\ds \map \arccos {\frac {\mathbf v \cdot \mathbf w} {\norm {\mathbf v} \norm {\mathbf w} } }$ Composite of Bijection with Inverse is Identity Mapping

$\blacksquare$