The vector product of two parallel vectors is:
The magnitude of the vector product is $ab\sin\theta$. If two vectors are parallel, $\theta = 0^\circ$ or $\theta = 180^\circ$. In both cases, $\sin\theta = 0$. Therefore, the magnitude of the vector product is zero, making it a zero vector. The text states: 'd(r $\times$ p)/dt = r $\times$ dp/dt + dr/dt $\times$ p. Now, the velocity of the particle is v = dr/dt and p = m v. Because of this d(r $\times$ p)/dt = r $\times$ dp/dt + v $\times$ mv. The term v $\times$ mv is zero as the vector product of two parallel vectors vanishes.'