Vector space with inner product/Orthogonal/Length symmetry/Remark

The following reasoning shows that the orthogonality, defined via the inner product, corresponds to the intuitively given orthogonality . For orthogonal vectors ${}u,v\in V$, we get that ${}v$ has the same distance to the points ${}u$ and ${}-u$. This holds because of

${}{\begin{aligned}\Vert {v-u}\Vert ^{2}&=\left\langle v-u,v-u\right\rangle \\&=\left\langle v,v\right\rangle +\left\langle u,u\right\rangle \\&=\left\langle v+u,v+u\right\rangle \\&=\Vert {v+u}\Vert ^{2}.\end{aligned}}$

The reverse statement holds as well; see exercise.