Would Rodrigues' rotation formula apply in this case?

Question

Suppose I am given two unit vectors starting from origin in three dimensions, $\vec{v} = (v_x, v_y, v_z)$ and $\vec{u}=(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta) $, which is just the spherical representation of the end point of the vector $\vec{u}$. Given the knowledge about the two vectors, I know that the angle between them is $$\cos\theta_0 = v_x\sin\theta\cos\phi + v_y\sin\theta\sin\phi + v_z\cos\theta~.$$ However, suppose I want to know all vectors that make angle $\theta_0$ with $\vec{v}$, i.e I want to know all the $(\theta, \phi)$. In this case, would Rodrigue's rotation formula apply?