The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point
\(\vec r'\) is a coordinate-independent, physical and geometric quantity. But,
in practice, you will need to know how to express this quantity in different
coordinate systems.
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Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point \(\vec
r'\) in rectangular coordinates.
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Show that this same distance written in cylindrical coordinates is:
\begin{equation}
\left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2}
\end{equation}
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Show that this same distance written in spherical coordinates is:
\begin{equation}
\left\vert\vec r -\vec r\,{}'\right\vert
=\sqrt{r'^2+r\,{}^2-2rr\,{}'
\left[\sin\theta\sin\theta\,{}'\cos(\phi-\phi\,{}')
+\cos\theta\cos\theta\,{}'\right]}
\end{equation}
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Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify
the previous two formulas.