As mentioned in every text book I own, the relative velocities of approach and separation are equal and opposite for (only) 1D collisions. For 2D collisions, only the component of velocities along the line of impact has to be considered. This makes sense since the (impulsive) force, ignoring friction, is always along the line of impact, and so must be the changes in momentum.
Here is the problem. In the derivation of this result, $\vec U_1-\vec U_2=-(\vec V_1- \vec V_2)$, at no point did I make the assumption that these velocities were 1 dimensional vectors. So shouldnt this formula apply to 2D and 3D collisions as well( without considering the velocities along the line of impact instead of the actual velocities)? Or is there some implicit assumption I already made in the Derivation that Restricts this?