Projection

Projections are extremely useful for finding shortest distances.

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

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

Norms

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

$\Vert {\text{perp}}_{\vec{v}}\vec{u}\Vert =\sqrt{\Vert \vec{u}{\Vert }^2-{\left(\frac{|\vec{u}\cdot \vec{v}|}{\Vert \vec{v}\Vert }\right)}^2}$

Related

• Dot Product