Exercise 23

Generalize to $\mathbb{R}^{n}$ the converse of the Pythagorean Theorem, that if $\vec{u}$ and $\vec{v}$ are perpendicular then $\lvert\vec{u}+\vec{v}\,\rvert^{2}=\lvert\vec{u}\,\rvert^{2}+\lvert\vec{v}\,\rvert^{2}$.