Momentum as a Vector

In two dimensions momentum is a vector along the velocity. It splits into components:

$$p_x = mv\cos\theta, \quad p_y = mv\sin\theta$$

The components of p are the sides of a right triangle, like any vector.

Back to the size: $p = \sqrt{p_x^2 + p_y^2}$. A component pointing the negative way is negative.

In a collision the total $\relax p_x$ and the total $\relax p_y$ each stay the same.