Exercise 6

Find the angle between the diagonal of the unit square in $\mathbb{R}^{2}$ and any one of the axes.

Find the angle between the diagonal of the unit cube in $\mathbb{R}^{3}$ and one of the axes.

Find the angle between the diagonal of the unit cube in $\mathbb{R}^{n}$ and one of the axes.

What is the limit, as $n$ goes to $\infty$, of the angle between the diagonal of the unit cube in $\mathbb{R}^{n}$ and any one of the axes?