So if I'm given an arbitrary unit vector $N$ and another vector $V$ defined in spherical coordinates $\theta$ (polar angle between $N$ and $V$) and $\phi$ (azimuthal angle) and $r = 1$. How do I convert vector $V$ into cartesian coordinates?
Now, I know that in general the conversion from spherical to cartesian is as follows:
$$x = r \sin \theta \cos \phi$$ $$y = r \sin \theta \sin \phi $$ $$z = r \cos \theta$$
However, since the angles $\theta$ and $\phi$ are defined respective to the vector $N$ and not the axes, the above conversion wouldn't work, yes? So how would I go about modifying the conversion?