Suppose I have a vector $\vec{a}$ in 3 dimensions. I realise that there is no unique solution for vectors a fixed angle $\alpha$ away from vector $\vec{a}$ - instead there is a cone of possible solutions.
Is there a way to parameterise the solution by some parameter than ranges over $0-2\pi$?
Any vector $\vec{b}$ an angle $\alpha$ away must satisfy,
$$ \text{cos } \alpha = \frac{\vec{a} \cdot \vec{b}}{|a||b|}$$
If we are dealing with unit vectors, then this becomes,
$$ \text{cos } \alpha = a_1b_1 +a_2b_2 +a_3b_3$$
Is it then possible to parameterise this expression for all possible vectors?
Thanks