$2$ vectors. decompose one vector in $2$ components.

Question

If I have two vectors $v$ and $q$ in $\mathbb{R}^{3}$, can I decompose the vector $v$ such that $v=a+n$, where $a$ is the component along the vector $q$ and $n$ is the component normal to $q$. What does it mean this statement? Can you help me, please?

Answer 1

Sure, the projection of $v$ onto $q$ is given by $v\cdot q / |q|$, this means $$v=\frac{v\cdot q}{|q|} + n\Rightarrow n = v-\frac{v\cdot q}{|q|}$$

Note that if $v$ is parallel to $q$ then $n=0$ as expected and if $v$ is perpendicular to $q$ then $n=v$ as expected.