I have a vector with spherical co-ordinates $(r_1,\theta_1,\phi_1)$, then I want this vector to be rotated by $\theta_2$ $\phi_2$ spherical angles but I cannot figure out how. I have tried using the rotational matrices
\begin{alignat}{1} R_z(\phi_2) &= \begin{bmatrix} \cos \phi_2 & -\sin \phi_2 & 0 \\[3pt] \sin \phi_2 & \cos \phi_2 & 0\\[3pt] 0 & 0 & 1\\ \end{bmatrix}\\[6pt] then\\ R_y(\theta_2) &= \begin{bmatrix} \cos \theta_2 & 0 & \sin \theta_2 \\[3pt] 0 & 1 & 0 \\[3pt] -\sin \theta_2 & 0 & \cos \theta_2 \\ \end{bmatrix} \\[6pt] \end{alignat}
but this does not reproduce \begin{alignat}{1} V_{new} &=|V| \begin{bmatrix} \sin\theta_2\cos\phi_2 \\[3pt] \sin\theta_2\sin\phi_2\\[3pt] \cos\theta_2\\ \end{bmatrix}\\[6pt] \end{alignat}
when multiplied by a unit vector
\begin{alignat}{1} V &=|V| \begin{bmatrix} 0 \\[3pt] 0\\[3pt] 1\\ \end{bmatrix}\\[6pt] \end{alignat}
Any suggestions how to do this?