Module 1 : A Crash Course in Vectors
Lecture 2 : Coordinate Systems
  Example 6 :
  Express unit vectors of spherical coordinate system in terms of unit vectors of cartesian system.
  Solution :
  From the point P drop a perpendicular on to the x-y plane. Denote $\vec{OP^\prime}$ by $\vec\rho$. The figure below shows the unit vectors in both the systems. By triangle law of vector addition,
 
\begin{displaymath}\vec r = \vec{OP} = \vec{OP'}+\vec{P'P} = r\sin\theta\hat\rho+ r\cos\theta\hat k \end{displaymath}
  However, $\hat\rho = \cos\phi\hat\imath + \sin\phi\hat\jmath$. Substituting this in the expression for $\vec r$, we get on dividing both sides by the magnitude of $r$
 
\begin{displaymath}\hat r = \sin\theta\cos\phi\hat\imath + \sin\theta\sin\phi\hat\jmath+ r\cos\theta\hat k\end{displaymath}
 


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