Today I had my first classical mechanics class (upper level undergraduate studies; I'm so glad!) and I'm already stuck with some "basic" math. I am however willing to put all my effort in solving all the problems.

I have to convert the versors $\displaystyle \hat i$, $\displaystyle \hat j$ and $\displaystyle \hat k$ into the versors of cylindrical and spherical coordinates, namely $\displaystyle \hat \rho$, $\displaystyle \hat \phi$, $\displaystyle \hat k$ for cylindrical coordinates and $\displaystyle \hat r$, $\displaystyle \hat \phi$ and $\displaystyle \hat \theta$ for spherical coordinates.

I've tried to write $\displaystyle \hat i$ as a linear (or non linear?) combination of $\displaystyle \hat \rho$ and $\displaystyle \hat \phi$ in cylindrical coordinates. But I always end up with $\displaystyle \hat i =\hat \rho \cos \phi$. Some of the students that could solve the problem ended up with $\displaystyle \hat i =\cos \phi \hat \rho - \sin \phi \hat \phi$. I'm totally clueless about how to reach the good result. How can I start?

I'm also asked to convert cylindrical versors into spherical ones and all other possible conversions.
I just need a push up in the right direction, I'm going to do all the arithmetic and everything involved in solving the problem, but I need a start-point.
Thank you very much in advance.