# Adding rotations onto a vector

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?

• I think the rotations should give $(r_1,\theta_1+\theta_2,\phi_1+\phi_2)$ acting on $(r_1,\theta_1,\phi_1)$ Dec 5, 2014 at 10:04
• Your proposed answer looks a lot like it's a rotation around $r$ only, see iac.ethz.ch/edu/courses/bachelor/veranstaltungen/… Dec 5, 2014 at 10:08

This is done with adding extra (service) dimension, which components are always 1 for vectors.
$\begin{bmatrix} a_1& a_2& a_3 & 1\\ \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ b_{41} & b_{42} & b_{43} & 1 \end{bmatrix} = \begin{bmatrix} a_1+b_{41}\\ a_2+b_{42}\\ a_3+b_{43}\\ 1 \end{bmatrix}$