Is $|(v,\frac{Pv}{||Pv||})|=||Pv||$ when $P$ is an orthogonal projection?

Question

Suppose $P$ is an $k \times k$ matrix that represents an orthogonal projection. Let $v$ be an $k \times 1$ vector. Let the operator $(\cdot,\cdot)$ represents the scalar product.

Does this equality holds $|(v,\frac{Pv}{||Pv||})|=||Pv||$ ? if so, how to prove it ? else, is there any condition under which this equality is correct ?

Answer 1

Hint:

$$(v,Pv) = (Pv+Rv,Pv) = (Pv,Pv)+(Rv,Pv) = \|Pv\|^2$$

where $Rv$ is a rejection (i.e. a vector orthogonal to $Pv$).