Why is this called the orthogonal projection of $u$ on $W$ if $proj_Wu$ is not orthogonal to $u$?
Solution 1:
As written in the text, $u = w_1 + w_2$ where $w_1 \in W$ and $w_2$ is orthogonal to $W$.
It is called an "orthogonal" projection because the difference $w_2 = u-w_1$ between $u$ and its projection is itself orthogonal to $W$.
(That it deserves to be called a projection is because $\mathrm{proj}_W(u) \in W$, and $\mathrm{proj}_W(\mathrm{proj}_W(u)) = \mathrm{proj}_W(u)$.)