$v$ is a vector. $n$ is normal vector for a (hyper)plane. What is the projection of $v$ onto that plane?
In three dimensions, let the projection be $x$. We have:
$$x\cdot n=0$$
$$(v-x)\cdot x=0$$
$$(v-x)\cdot n=|v-x|$$
These three equations might simply to:
$$v\cdot n=|v-x|$$
$$v\cdot x-x^2=0$$
I don't know how to proceed..
Seems like: $$x=v-(n\cdot v)n?$$
Best Answer
Hint:
The projection of $v$ onto a line directed by $n$ is $$p_n(v)=\frac{v\cdot n}{n\cdot n}\, n.$$