In https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula
it distinguishes angles $\theta$ and $\alpha$ and says "the two angles $\theta$ and $\alpha$ are not necessarily equal"
To my knowledge, I think they should be same. When we use dot product and cross product, we use the same angle, that is the angle between two vectors. Why are they not the same?
What they mean is that the axis of rotation $\hat{\bf k}$ can be obtained via the product of two other vector ${\bf a}$ and ${\bf b}$
$$ \hat{\bf k} = \frac{{\bf a}\times {\bf b}}{|{\bf a}\times {\bf b}|} $$
it just happens that the angle between these two vectors is $\alpha$