Vector Cosine

The Cauchy-Schwarz Inequality can be written

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

In the case of real vectors $\underline{x},\underline{y}$, we can always find a real number $\theta$ which satisfies

$$\cos(\theta) \isdef \frac{\left<\underline{x},\underline{y}\right>}{\Vert\underline{x}\Vert\cdot\Vert\underline{y}\Vert}.$$

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