> [!Definition] Definition (Angle) > Let $\underline{v},\underline{w}\in \mathbb{R}^{n}\setminus \{ 0 \}$ be non [[Real Zero Vector|zero]] [[Real n-Space|real vectors]]. We define the angle between $\underline{v}$ and $\underline{w}$, denoted $\angle \underline{vw}$, in terms of their [[Dot Product in Real n-Space|dot product]] and [[Euclidean Norm|length]] as follows: $\angle \underline{vw} = \cos^{-1} \left( \frac{\underline{v} \cdot \underline{w}}{||\underline{v} ||\, ||w ||} \right)$ where $\cos^{-1}$ denotes [[Arccos|arccos]] so that $\angle \underline{v} \underline{w}$ lies in the [[Real intervals|real interval]] $[0,\pi].$ > **Note**: [[Cauchy-Schwartz inequality]] asserts that ${ (\underline{v} \cdot \underline{w} )}/{(||v|| \, ||w||)}$ lies in the interval $[-1,1].$