Let $m \in \mathbb{N}^{\star}$ and $(u_1, \ldots, u_m)$ be a family of vectors of $\mathbb{R}^n$. We denote
$$C = \left\{ \sum_{i=1}^{m} \mu_i u_i, \mu_i \geqslant 0 \; \forall i \in \llbracket 1, m \rrbracket \right\}.$$
We consider a vector $v \in \mathbb{R}^n \setminus C$.\\
a. Show that $\langle P_C(v), P_C(v) - v \rangle = 0$.\\
b. We set $w = P_C(v) - v$. Show that $\langle v, w \rangle < 0$ and $\langle u_i, w \rangle \geqslant 0$ for all $i \in \llbracket 1, m \rrbracket$.