In an angle-free representation, the changed velocities are computed using the centers x 1 and x 2 at the time of contact as v 1 ′ = v 1 − − 2 m 2 m 1 + m 2 ⟨ ⟨ v 1 − − v 2 , x 1 − − x 2 ⟩ ⟩ ‖ ‖ x 1 − − x 2 ‖ ‖ 2 ( x 1 − − x 2 ) , v 2 ′ = v 2 − − 2 m 1 m 1 + m 2 ⟨ ⟨ v 2 − − v 1 , x 2 − − x 1 ⟩ ⟩ ‖ ‖ x 2 − − x 1 ‖ ‖ 2 ( x 2 − − x 1 ) where the angle brackets indicate the inner product (or dot product) of two vectors.