The position vector field is easier to write algebraically in rectangular coordinates than it is to think about: \begin{equation} \rr=x\,\ii+y\,\jj+z\,\kk \label{position} \end{equation}

But what about in curvilinear coordinates? If you try to write the position vector $\rr(P)$ for a particular point $P$ in spherical coordinates, and you think of the tail of the position vector as “attached” to the origin, then you have a problem. It is not clear which $\rhat$, $\that$, and $\phat$ you should use. The resolution to this dilemma, of course, is to think of the tail as “attached” to the point $P$ and to use the $\rhat$, $\that$, and $\phat$ at $P$. We can use a subscript $P$ to denote this particular choice of basis vectors $\rhat_P$, $\that_P$, and $\phat_P$. If you are trying to use the position vector to tell someone how to get to $P$ from the origin and you try to use curvilinear coordinate basis vectors, it turns out to be tricky.

Naive pattern matching with ($\ref{position}$) might lead you to believe that the position vector in spherical coordinates is given by: \begin{equation} \rr(P)=r\,\rhat_P+\theta\,\that_P+\phi\,\phat_P\qquad\qquad\hbox{(incorrect)} \end{equation} But, if you try to follow this equation as a literal set of instructions, then the instructions say “first travel the distance $r$ in the $\rhat_P$ direction (which already gets you to where you want to go) and then from there, travel the distance $\theta$ (which is an angle, not a distance) in the $\that_P$ direction and then from there, travel the distance $\phi$ (which is also an angle, not a distance) in the $\phat$ direction.” These directions are clearly incorrect; we should have stopped after the first step. In spherical coordinates, the position vector is given by: \begin{equation} \rr(P)=r\,\rhat_P\qquad\qquad\hbox{(correct)} \end{equation}

You should try to use a similar process to find the position vector in cylindrical coordinates. If you are trying to use the position vector to tell someone how to get to $P$ from the origin and you try to use curvilinear coordinate basis vectors, then they have to know where $P$ is in order to understand your description in terms of the basis vectors, $\rhat_P$, $\that_P$, and $\phat_P$, at $P$!