Module 1 : A Crash Course in Vectors
Lecture 2 : Coordinate Systems
  The unit vector $\hat\theta$ is perpendicular to OP in the direction of increasing $\theta$. The angle that $\hat\theta$ makes with the positive z-axis is $\pi/2+\theta$.
  Hence,
 
\begin{displaymath}\hat\theta = \cos(\frac{\pi}{2}+\theta)\hat k + \sin(\frac{\pi}{2}+\theta)\hat\rho = -\sin\theta\hat k + \cos\theta\hat\rho\end{displaymath}
  Substituting for $\rho$, we get
 
\begin{displaymath}\hat\theta = \cos\theta\cos\phi\hat\imath + \cos\theta\sin\phi\hat\jmath- \sin\theta\hat k\end{displaymath}
  The azimuthal unit vector $\vec\phi$ is in the direction of increasing angle $\phi$. It is perpendicular to $\vec\rho$ and has no z- component. It can be easily seen that
 
\begin{displaymath}\hat\phi = -\sin\phi\hat\imath + \cos\phi\hat\jmath\end{displaymath}
 


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