For vector v in the span of orthogonal basis v_1, ..v_{n}:
\begin{equation} v = c_1 v_1 + \dots + c_{n} v_{n} \end{equation}
we can write:
\begin{equation} c_{j} = \frac{v \cdot v_{j}}{ v_{j} \cdot v_{j}} \end{equation}
Proof:
\begin{equation} \langle v, v_{j} \rangle = c_{n} \langle v_{1}, v_{j} \rangle \dots \end{equation}
which is 0 for all cases that’s not \langle v_{j}, v_{j} \rangle as the v are orthogonal, and \mid v_{j} \mid^{2} for the case where it is. Hence, we see that:
\begin{equation} \langle v, v_{j} \rangle = c_{j} \mid v_{j}\mid^{2} \end{equation}
Which gives:
\begin{equation} c_{j} = \frac{\langle v,v_{j} \rangle}{\mid v_{j}\mid^{2}} \end{equation}
as desired.