Prove: $\langle p, v\rangle = \|p\|^2$

Question

Need some help proving the following:

Let $V$ be a Inner product space over field $\mathbb F$ (real or complex). Let $U$ be a sub-space of $V$. Let $v$ be a vector from $V$, and let $p$ be the projection of $v$ onto $U$.

Prove: $\langle p, v\rangle = \|p\|^2$

Thinking about using the linearity of the inner product but get stuck on the way.

Thank you

Answer 1

Hint: Assuming that you are talking about the orthogonal projection, use the fact that $p=(p-v)+v$ and that $\|p\|^2=\langle p,p\rangle$.