Angle Between Vectors in Terms of Dot Product

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

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


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


\(\ds \norm {\mathbf v} \norm {\mathbf w} \cos \theta\) \(=\) \(\ds \mathbf v \cdot \mathbf w\) 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


Also see