Projection

Projections are extremely useful for finding shortest distances.

${\text{proj}}_{\vec{v}}\vec{u}=\frac{\vec{u}\cdot \vec{v}}{\Vert \vec{v}{\Vert }^2}\vec{v}$

${\text{perp}}_{\vec{v}}\vec{u}=\vec{u}-{\text{proj}}_{\vec{v}}\vec{u}$