The distance \(\left\vert\vec r \vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point
\(\vec r'\) is a coordinateindependent, physical and geometric quantity. But,
in practice, you will need to know how to express this quantity in different
coordinate systems.

Find the distance \(\left\vert\vec r \vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point \(\vec
r'\) in rectangular coordinates.

Show that this same distance written in cylindrical coordinates is:
\begin{equation}
\left\vec r \vec r\,{}'\right =\sqrt{s^2+s\,{}'^22ss\,{}'\cos(\phi\phi\,{}') +(zz\,{}')^2}
\end{equation}

Show that this same distance written in spherical coordinates is:
\begin{equation}
\left\vert\vec r \vec r\,{}'\right\vert
=\sqrt{r'^2+r\,{}^22rr\,{}'
\left[\sin\theta\sin\theta\,{}'\cos(\phi\phi\,{}')
+\cos\theta\cos\theta\,{}'\right]}
\end{equation}

Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)\(y\) plane. Simplify
the previous two formulas.