 Solution

Let $a=\Vert {\Pi }_A\xi \Vert$ and $b=\Vert {\Pi }_{A^{\perp }}\xi \Vert$. Since $a^2+b^2=1$, choose $\theta \in [0,\pi /2]$ with $a=\sin \theta$, $b=\cos \theta$. Normalizing the two nonzero projections givesAt an endpoint, the unused normalized vector may be chosen arbitrarily in its subspace.