I have a unit vector $\hat{a}$ and a unit vector $\hat{k}=\{\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\}$. Now I can rotate $\hat{k}$ such that it aligns with the z-axis, i.e. $\hat{k} \rightarrow \hat{k}'=\{0,0,1\}$. Is there a way to relate the dot product $\hat{k}.\hat{a}$ and $\hat{k}'.\hat{a}$ given the two angles $\theta,\phi$ involved? I can find the Euler angles for the rotation but not sure how to proceed from there.
A similar question was explored before: Find the angle between two vectors after rotating it.. But I was wondering if I can parametrize the relation in terms of just the angles involved in the rotation.
Any help would be appreciated. TIA!