The wikipedia article Angles between flats discusses the principal angles between two subspaces of $\mathbb{R}^n$. It states, "if the largest angle is $π / 2$, there is at least one vector in one subspace perpendicular to the other subspace."
How can one find that vector?
Also, I'm having trouble following the definitions laid out in the article. Does anyone know of a text or other reference that covers angles between subspaces?
Let $V$, $W$ be two subspaces of $\mathbb R^n$ which are perpendicular (i.e. there is a vector from $W$, which projection on $V$ is zero, so it is perpendicular).
Suppose $A$ is the projection matrix onto $V$. Then, every vector perpendicular to $V$ is an element of null space of $A$. Furthermore, $W$ can be described as a set of solutions of system of linear equations. To get your vector, just solve the system of equations.