[math] \newcommand{\mathds}{\mathbb}[/math]

Develop a theory of angles between the vectors [math]x,y\in\mathbb R^N[/math], by using the well-known formula

[[math]] \lt x,y \gt =||x||\cdot||y||\cdot\cos t [[/math]]

that you should by the way fully understand first, in [math]N=2[/math] dimensions.