 Solution

The assumption $v\notin W^{\perp }$ means $p=Pv\ne 0$. For every unit vector $w\in W$, orthogonality and Cauchy–Schwarz give $⟨v,w⟩=⟨p,w⟩\le \Vert p\Vert$. Equality is attained atEquality in the real Cauchy–Schwarz inequality requires $w$ to be a scalar multiple of $p$, and attaining the positive supremum fixes its sign; the unit norm fixes its magnitude. This proves uniqueness.