There is probably something dreadfully simple I am missing here, but in the derivation of the speeds after a collision of any elasticity, there is a step that involves going from:
$$C_r = \frac{|v_2 - v_1|}{|u_1 - u_2|}$$
... to this:
$$v_2 = v_1 + C_r(u_1 - u_2)$$
... where $C_r$ is the coefficient of restitution, $u_1$ and $u_2$ are the initial velocities of two particles and $v_1$ and $v_2$ are their final velocities.
How is it legal to rearrange the equation as seen above? What happened to the magnitudes that had been taken?
Thank you.
(here is an example of this derivation)