Definition:Angle Between Vectors in Inner Product Space

Definition

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $v,w \in V$ be such that $\size v, \size w \ne 0$, where $\size {\, \cdot \,}$ denotes the inner product norm.

Then the real number $\theta \in \closedint 0 \pi$ is called the angle (between $v$ and $w$) if and only if:

$\map \cos \theta = \dfrac {\innerprod v w} {\size v \size w}$

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions