Projecting onto a subspace given the quotient space?

Question

I have a hyperplane in ${\rm I\! R}^n$ described by a normal vector $\mathbf{v}$. I would like to project some point $x \in {\rm I\! R}^n$ onto the subspace ${\rm I\! R}^n / \mathbf{v}$. If I had a basis for ${\rm I\! R}^n/\mathbf{v}$, I could accomplish such a thing by simply iteratively projecting onto those basis vectors. Not having them, it seems a waste to go through all the effort of finding such a basis, and then performing the projections. Is there a way to obtain such a projection directly from the normal vector $\mathbf{v}$ and a point $x$?

Answer 1

Do you want to project a general point $x\in\Bbb R^n$ to a point in the plane perpendicular to $v$?

If $v$ is a unit vector, that's easy: the point $x$ must go to $y=x-\lambda v$ for some real $\lambda$, and then $v\cdot y=0$. That means that $v\cdot x-\lambda =0$, so $y=x-(v\cdot x)v$.

And if $v$ isn't a unit vector? You could scale it first....