Answer
see the proof below.
Work Step by Step
Let $w=u-\operatorname{proj}_{{v}} {u}$, then $w=u-\frac{\langle{u}, {v}\rangle}{\langle{v}, {v}\rangle} {v}$
\begin{aligned}\langle{w}, {v}\rangle &=\left\langle{v}, {u}-\frac{\langle{u}, {v}\rangle}{\langle{v}, {v}\rangle} {v}\right\rangle \\ &=\langle{v}, {u}\rangle-\frac{\langle{u}, {v}\rangle}{\langle{v}, {v}\rangle}\langle{v}, {v}\rangle \\ &=\langle{u}, {v}\rangle-\langle{u}, {v}\rangle \\ &= 0 \end{aligned}.
This ends the proof.