title: Converting from spherical coordinates to cartesian around arbitrary vector N
body: <p>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? </p>
  
  <p>Now, I know that in general the conversion from spherical to cartesian is as follows:</p>
  
  <pre><code>x = r * sin theta * cos phi 
  y = r * sin theta * sin phi 
  z = r * cos theta
  </code></pre>
  
  <p>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?</p>