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{\bf Question}

From the relations \begin{eqnarray*} {\bf e}_r & = & {\bf i} \sin
\theta \cos \phi + {\bf j} \sin \theta \sin \phi + {\bf k} \cos
\theta \\ {\bf e}_\phi & = & - {\bf i} \sin \phi + {\bf j} \cos
\phi \\ {\bf e}_\theta & = & {\bf i} \cos \theta \cos \phi + {\bf
j} \cos \theta \sin \phi - {\bf k} \sin \theta \end{eqnarray*}

verify that ${\bf e}_r,{\bf e}_{\phi}, {\bf e}_{\theta},$ the
basis vectors of spherical polar coordinates, form an orthogonal
set, i.e. that they each have unit length and are mutually
orthogonal.

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{\bf Answer}

Showing that the vectors have unit length:
\begin{eqnarray*}
|{\bf e}_r| & = & (\sin \theta \cos \phi)^2 + (\sin \theta \sin
\phi)^2 + (\cos \theta)^2 \\ & = & \sin ^2 \theta \cos^2 \phi +
\sin^2 \theta \sin^2 \phi + \cos^2 \theta \\ & = & \sin ^2 \theta
(\cos^2 \phi + \sin^2 \phi) + \cos^2 \theta \\ & = & \sin ^2
\theta + \cos^2 \theta \\ & = & 1 \\ \\ |{\bf e}_\phi| & = & (\sin
\phi)^2 + (\cos \phi)^2 \\ & = & 1 \\ \\ |{\bf e}_\theta| & = &
(\cos \theta \cos \phi)^2 + (\cos \theta \sin \phi)^2 + (\sin
\phi)^2 \\ & = & \cos^2 \theta \cos^2 \phi + \cos^2 \theta \sin^2
\phi + \sin^2 \phi \\ & = & \cos^2 \theta (\cos^2 \phi + \sin^2
\phi) + \sin^2 \phi \\ & = & \cos^2 \theta  + \sin^2 \phi \\ & = &
1 \end{eqnarray*}

Showing they the vectors are mutually orthogonal:

\begin{eqnarray*} {\bf e}_r \cdot {\bf e}_\phi & = & -\sin \theta
\cos \phi \sin \phi + \sin \theta \cos \phi \sin \phi \\ & = & 0
\\ \\ {\bf e}_r \cdot {\bf e}_\theta & = & \cos \theta \cos^2 \phi
\sin \theta + \sin \theta \sin^2 \phi \cos \theta - \cos \theta
\sin \theta \\ & = & \cos \theta \sin \theta(\sin^2 \phi + \cos ^2
\phi) - \cos \theta \sin \theta \\ & = & 0 \\ \\ {\bf e}_\phi
\cdot {\bf e}_\theta & = & -\cos \theta \cos \phi \sin \phi + \cos
\phi \cos \phi \sin \phi \\ & = & 0 \end{eqnarray*}



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