Why is this called the orthogonal projection of $u$ on $W$ if $proj_Wu$ is not orthogonal to $u$?

Question

Why is this called the orthogonal projection of $\mathbf u$ on $W$ if $proj_W \mathbf u$ is not orthogonal to $\mathbf u$?

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Answer 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 $\mathrm{proj}_W(\mathrm{proj}_W(u)) = \mathrm{proj}_W(u)$.)

Answer 2

Simply because $(\operatorname{proj}_W \mathbf u - \mathbf u)$ is orthogonal to W, i.e. its projection on $W$ is a zero-length vector ("orthogonal" to it). So if you make an analogy with 3D space, to make this projection you drop a perpendicular on the "plane" $W$.