Oblique impacts FM1

Hello! I have a question about conservation of momentum in oblique impacts of smooth spheres. I've learnt that momentum is conserved along the line of centres. Is it only conserved along this line? Why when dealing with vectors is it okay and correct to use both the i and j components? I think my logic has gone wrong somewhere, thank you.

Momentum is conserved along any line. When two spheres collide the total momentum will remain the same before/after collision.

The point here is that during a collision, you should be describing the problem with reference to the line of centres and the perpendicular to the line of centres. You decompose your problem into these two directions and treat them separately. In the simplest of cases, these lines are described by your i and j components.

Using this description, the velocity of the spheres along the line of centres changes but momentum is conserved so we can relate these velocities before/after. The velocity along the perpendicular line remains the same, and this also implies conservation of momentum along this line, and since these velocities are completely the same before/after it is what allows us to ignore this direction in the problem, making it simpler to find the before/after velocities.