Why isn't projection of $\mathbf{v}$ onto $\mathbf{u}$ defined so that it's perpendicular to the straight line segment connecting $\mathbf{u, v}$? Kuldeep Singh. Linear Algebra: Step by Step
(2013). p 312
Why isn't $\mathbf{proj_u v}$ defined as the red vector below with $\mathbf{v} \perp \mathbf{d}?$ I already know, but can't intuit why, $\mathbf{proj_u v}$ is defined as the green vector below with $\mathbf{p} \perp \mathbf{u}$.




By definition, projection of $\vec v$ onto $\vec u$ is vector $\vec w$ parallel to $\vec u$ such that
$$\vec v=\vec w+\vec p$$
therefore, as an intuition, $\vec v$ is the longest vector in the triple.
Indeed, in your alternative sketch, $\vec v$ represents the projection of the red vector onto $\vec v$ direction.
