Adding rotations onto a vector

Question

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?

Answer 1

The rotation matrices you used only apply to rotations around cartesian axes (x,y,x). You cannot generalize it to spherical coordinates in the way you did. See page 4 here for the rotation matrix in spherical coordinates.

Answer 2

Looks like you wish to express matrix addition with matrix multiplication.

This is done with adding extra (service) dimension, which components are always 1 for vectors.

$\begin{bmatrix} a_1& a_2& a_3 & 1\\ \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ b_{41} & b_{42} & b_{43} & 1 \end{bmatrix} = \begin{bmatrix} a_1+b_{41}\\ a_2+b_{42}\\ a_3+b_{43}\\ 1 \end{bmatrix} $

You can ignore this forth dimension in all cases except calculating the rotation.