Indicating with and the velocities of the particles after impact, the conservation of momentum and kinetics energy leads to the equations...

(1)

... which is equivalent to the scalar equations system...

(2)

We have now three equation and four unknown variables, so that we need another equation. This equation is supplied by the supplementar condition that indicates that and are orthogonal, so that is...

(3)

... and the problem can be solved...

My question [for experts...] is : the condition (3), necessary to have a well defined problem, isn't in some way arbitrary?...

Kind regards