Even though I have worked with cylindrical and spherical coordinates before, I am now getting confused when getting used to their unit vectors.
E.g. the vector $(1, \frac{\pi}{4},2) $ in cylindrical coordinates seems clear at first.
However which unit vectors do we actually use to get to the point? It seems to me to be neither the cartesian unit vectors, nor the cylindrical ones since $ \vec{e_{\theta}} $ depends on the current point and is obviously straight and not the arc length/angle in rad. Can we not write it in a sum of coefficients and unit vectors like $ \vec{r}= \sum_{i} a_i \vec{e_i} $? I have seen that $a_{\theta} =0 $ is set to zero which does make sense graphically, but then where does $\frac{\pi}{4} $ go?
I have the same problem with spherical coordinates but I think this will settle once I get cylindrical ones. So I would appreciate if someone could explain it, since I can't wrap my head around it by myself.