Consider two vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^{n}$. Show that a may be written as the sum of two vectors: one parallel (or anti-parallel) to $\mathbf{b}$ and the other perpendicular to $\mathbf{b}$. By setting the former equal to $\cos \theta|\mathbf{a}| \hat{\mathbf{b}}$, where $\hat{\mathbf{b}}$ is a unit vector along $\mathbf{b}$, show that

$$\cos \theta=\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$$

Explain why this is a sensible definition of the angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$.

Consider the $2^{n}$ vertices of a cube of side 2 in $\mathbb{R}^{n}$, centered on the origin. Each vertex is joined by a straight line through the origin to another vertex: the lines are the $2^{n-1}$ diagonals of the cube. Show that no two diagonals can be perpendicular if $n$ is odd.

For $n=4$, what is the greatest number of mutually perpendicular diagonals? List all the possible angles between the diagonals.