Answer
False
Work Step by Step
We have the notation for the inner product
$P=\frac{}{||v||^2}v$
then
$\\
={||v||^2}v>\\
=-{||v||^2}v>\\
=-\frac{}{||v||^2}\\
=||w||^2-\frac{}{||v||^2}$
If we let $w=(1,1)$ and $v=(1,-1)$
then it becomes $||w||^2=2\\
||v||^2=2\\
^2=0\\
\rightarrow =2-0=2 \ne 0$
Hence, $w-P(w,v)$ is not orthogonal to $w$
