Computing velocity after elastic collision of two balls

[Christopher Munroe]

I am a programmer trying to simulate some 2D balls bouncing about and colliding with each other. I have both the ball's velocity components before the collision and I am trying to solve for them. I went to wikipedia to find a formula to satisfy my needs and ran into this section: https://en.wikipedia.org/wiki/Elastic_collision#Two-dimensional_collision_with_two_moving_objects
I am attempting to use an angle free representation as it is more convenient for my current design:

I am somewhat confused by it. The passage states the changed velocities are computed using the centers x1 and x2 at the time of contact. Isn't x1 and x2 scalars, I'm not sure what x1 and x2 are to be quite honest. If they are scalars, then we cannot perform a dot product. I am sorry if this is trivial, I appreciate any guidance.

 

[DrClaude]

They are vectors, with the two components of $\mathbf{x}_i$ corresponding to $(x_i,y_i)$.