Finding position vector in general local basis

How do you derive the position vector in a general local basis?

For example, in spherical coordinates, it's ##\vec r =r \hat {\mathbf e_r}##, not an expression that involves that involves the vectors ## {\hat {\mathbf e_{\theta}}}## and ## \hat {{\mathbf e_{\phi}}}##. But how would you show this?

This follows from the definition of the coordinate system. For example, in spherical polar coordinates, [itex]r[/itex] is by definition the distance of a point from the origin and [itex]\mathbf{e}_r(\theta,\phi)[/itex] is the unit vector in the direction of that point. Hence [itex]\mathbf{r} = r\mathbf{e}_r(\theta,\phi)[/itex].

Otherwise, if you have a global Cartesian basis then you can express the cartesian coordinates and basis vectors in terms of the curvilinear coordinates and basis vectors.