Projections

$${\text{proj}}_{\vec{a}}\vec{b}=\frac{\vec{a}\cdot \vec{b}}{\Vert \vec{a}{\Vert }^2}\vec{a}=(\hat{a}\cdot \vec{b})\hat{a}$$ $${\text{comp}}_{\vec{a}}\vec{b}=\Vert {\text{proj}}_{\vec{a}}\vec{b}\Vert =\frac{\vec{a}\cdot \vec{b}}{\Vert \vec{a}\Vert }$$

A vector is $=\vec{0}$ if its projection on two non parallel vectors is also $\vec{0}$

Expressing a Vector in Terms of Any Orthogonal Set

$$\vec{a}=pro{j}_{\overrightarrow{u_1}}\vec{a}+pro{j}_{\overrightarrow{u_2}}\vec{a}+pro{j}_{\overrightarrow{u_3}}\vec{a}$$