In an application test that I had to do for a job recently, I was asked the following question (I quote):
“Given three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. Compute the rotation (the angle) around $\mathbf{a}$ that brings $\mathbf{c}$ onto the plane defined by $\mathbf{a}$ and $\mathbf{b}$.
Hint: Solve the problem using vector operations.
Okay, to be honest, the hint didn't help. I cannot find a simple solution to this problem. Any ideas?
PS: If the question should not be all too clear, that would be part of the challenge, I guess.
Provided that the plane is actually defined by vectors $a$ and $b$ - find a vector product of these two, then compute scalar (dot) product of the found vector and the vector $c$, then divide the result by both norms, and you'll get a cosine of the angle, complementary to an angle you need.
The vector product of two vectors is perpendicular to the plane, defined by them.