Angle Between Vectors in Terms of Dot Product

Theorem

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

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

where:

$\mathbf v \cdot \mathbf w$ represents the dot product of $\mathbf v$ and $\mathbf w$

$\norm {\, \cdot \,}$ represents vector length

$\arccos$ represents arccosine.

$\ds \norm {\mathbf v} \norm {\mathbf w} \cos \theta$

$=$

$\ds \mathbf v \cdot \mathbf w$

$\ds \cos \theta$

$=$

$\ds \frac {\mathbf v \cdot \mathbf w} {\norm {\mathbf v} \norm {\mathbf w} }$

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 $0 \le \theta \le \pi$ $\implies -1 \le \cos \theta \le 1$

$\ds \theta$

$=$

$\ds \map \arccos {\frac {\mathbf v \cdot \mathbf w} {\norm {\mathbf v} \norm {\mathbf w} } }$

$\blacksquare$