projection vector

The projection vector of $\vec{u}$ onto $\vec{v}$ is a vector in $\vec{v}$ direction with a magnitude of $\vec{u}$‘s $\vec{v}$-component.

Its magnitude is

$$\left|{\mathrm{proj}}_{\vec{v}}\vec{u}\right|=\frac{\vec{u}\cdot \vec{v}}{\left|\vec{v}\right|}=\frac{\vec{u}\cdot \vec{v}}{\vec{v}\cdot \vec{v}}$$

To make it a vector in $\vec{v}$‘s direction, multiply it by $\frac{\vec{v}}{|\vec{v}|}$, which is $\vec{v}$‘s direction vector.