Vector Cosine

The Cauchy-Schwarz Inequality can be written

$$\frac{\left\vert\left<\underline{u},\underline{v}\right>\right\vert}{\Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert} \leq 1.$$

In the case of real vectors $\underline{u},\underline{v}$, we can always find a real number $\theta\in[0,\pi]$ which satisfies

$$\zbox {\cos(\theta) \isdef \frac{\left<\underline{u},\underline{v}\right>}{\Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert}.}$$

We thus interpret $\theta$ as the angle between two vectors in ${\bf R}^N$.