I always have trouble when working in spherical coordinates.
Given $|V|$ and the cartesian components $V_x,V_y,V_z$ I want to find the spherical components $V_r,V_\theta,V_\phi$. As far as I understand $V_r=|V|$, but how do I find the other components?
Trivially, $\theta=\cos^{-1}{V_z \over \sqrt{V_x^2+V_y^2}}$ and $\phi=\tan^{-1}{V_y \over V_x} (+\pi)$ but these are not `components of a vector' as you can't add two vectors by adding these 'components', or multiply them by a scalar.