A symmetry identity involving norms

Question

Let $x,y\ne 0$ in $\mathbb R^n$. Give a geometrical interpretation of the identity

$\Big \Vert\frac {x}{\Vert x\Vert} -\Vert x\Vert y\Big \Vert = \Big \Vert\frac {y}{\Vert y\Vert} -\Vert y\Vert x\Big \Vert $.

Here $\Vert\cdot\Vert $ denotes the Euclidean norm. I can prove the identity algebraically; both sides are $1-2\langle x,y\rangle +\Vert x\Vert^2+\Vert y\Vert^2$. What is a geometrical interpretation?

Answer 1

Let $u=x/\|x\|,\ v=y/\|y\|$ and $p=\|x\|\|y\|$. The identity can then be rewritten as $$ \|u-pv\|=\|v-pu\|. $$ It's true simply because there exists a linear isometry $Q$ such that $Qu=v$ and $Qv=u$.