I'll try to help until someone else gets to it.
I get that you have three vectors that are unit vectors. As your final answer though you want to have something like x=?, y=?, z=? in terms of a,b and c?
Are x,y and z just unit vectors themselves? I'm not quite following what you want to solve. It sounds like you are just stuck with the algebra, which I might be able to help with, but I'm lost conceptually to what you meant.
Welcome to MHF by the way!