Question

What is the projection of a vector $u$ onto a nonzero vector $v$?

Answer 1

The projection of a vector $u$ onto a nonzero vector $v$ is the vector denoted and defined by
$$
\operatorname{proj}(u, v)=\frac{u \cdot v}{\|v\|^{2}} v=\frac{u \cdot v}{v \cdot v} v
$$

This shows that this agrees with the usual notion of vector projection in physics.