"Find a unit vector parallel to $\displaystyle i + rj + 2k$ and perpendicular to $\displaystyle 2i + 2j - k$."

If I call the unit vector U(a,b,c) , then kU (k=some constant) is equal to the parallel vector. U dotted with the second vector should equal zero to be perpendicular to each other.

The big problem arises: there are so many equations and unknowns, that everything becomes impossible! Can anyone help me figure out the smartes way to solve this question?