Suppose I have two vectors on a unit sphere in spherical coordinates
$$ v_1 = [\theta_1,\phi_1,1]$$
$$ v_2 = [\theta_2,\phi_2,1]$$
I want to find a rotation matrix $R$ such that:
$$v_1' = R * v_1 = [0,0,1] $$
$$v_2' = R * v_2 = [\Theta,0,1] $$
Is it possible to construct a rotation matrix in spherical coordinates without converting to cartesian coordinates first?
I found this question where one of the answers suggests a rotation matrix to rotate by $\alpha$ in one axis is:
$R(\alpha) = \left( \begin{array}{ccc} 1 & 0 & \alpha \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) $
But I'm struggling to adapt this for three-axis rotation.