Proposition (Orthogonal decomposition)

Let $V$ be an Inner Product Space, with $v,w\in V$ and $w\not=0$. If we set $$c=\frac{\langle v, w \rangle }{\lVert w \rVert ^{2}}$$ and $u=v-cw$, then $$v=u+cw$$ and $u$ and $w$ are Orthogonal $$\langle u, w \rangle =0$$