Let $c \in \mathbb{R}$, $\alpha =(\alpha_1, \ldots, \alpha_n )$ and $L= \{ x \in \mathbb{R}^n : \langle \alpha,x\rangle=c \}$ (hyperplane). Show that for all $a \in \mathbb{R}^n$, $\mathrm{d}(a,L)=\frac{ |\langle a, \alpha \rangle-c|} {\| \alpha\|}$.
We know from the projection theorem that for all $x \in \mathbb{R}^n$, $\| x- \pi_L(x)\|=\inf_{z \in L} \| x-z \|=\mathrm{d}(a,L) $
How I can write $\pi_L(x)$ ?