\begin{proof}
$$\left \langle u,v_i \right \rangle=\left \langle \alpha_1v_1+\cdots+\alpha_nv_n,v_i \right \rangle
=\alpha_1\underbrace{\left \langle v_1,v_i \right \rangle}_{=0}
+\cdots+
\alpha_i\underbrace{\left \langle v_i,v_i \right \rangle}_{=1}
+\cdots+
\alpha_n\underbrace{\left \langle v_n,v_i \right \rangle}_{=0}=\alpha_i$$
\end{proof}