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 \hat i, \hat j and \hat k into the versors of cylindrical and spherical coordinates, namely \hat \rho, \hat \phi, \hat k for cylindrical coordinates and \hat r, \hat \phi and \hat \theta for spherical coordinates.

I've tried to write \hat i as a linear (or non linear?) combination of \hat \rho and \hat \phi in cylindrical coordinates. But I always end up with \hat i =\hat \rho \cos \phi. Some of the students that could solve the problem ended up with \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.