Does this relation about direction of particles make sense?

Question

Maybe I've just stared at this statement too long and I've missed something obvious. Nevertheless, here's the problem: Landau-Lifshitz vol. 1§16, problem 1.

Consider (classical) collision of two particles in center of mass coordinates. Before the collision, the particles are just traveling towards each other and in CM coordinates then the direction of the velocities of two particles should be opposite to each other, i.e. $$ \theta_1 = \theta_2 + \pi , $$ where $\theta_i$ is the angle between the velocity of particle $i$ and the coordinate axis.

However, in the solution in Landau-Lifshitz, they state that "In the C system, the corresponding angles are related by $\theta_1 = \pi - \theta_2$."

Is there a mistake in L-L? If not, can you explain me the relation above?

Answer 1

The difference comes from the picture--- the $\theta_1$ and $\theta_2$ in the original statement are both relative to the positive x-axis, while in the solution $\theta_2$ is the final angle relative to the initial velocity of the corresponding particle, so if the velocity is along the x-axis, $\theta_1$ is the angle relative to the x-axis, and $\theta_2$ is relative to the minus x axis.