The angular momentum of a particle of mass m at a position given by  is   =  vector r  X vector p  where vector p indicates the linear momentum (mv)

The above vector representation represents cross product of vector r and vector p. Hence, there can be two cases when  could be zero

Case 1 – When vector r is itself zero

Case 2 – When the angle between  vector r and vector p  is zero as the cross product of vector r and vector p gives rp sinθ = rp sin 0 = 0 as sin 0 is 0

Considering the same in graphical manner for a free fall object will help us to understand better.

Let the vertical Y axis be the direction of the free fall body. The angular momentum at a point P with respect to the free fall body of mass m will be is

               =    X  (the y component of  which is vector p_y )

Now let us change the choice of the point P to point Q so that the angle between  and  is zero. Now as θ is zero, sin θ is zero ; Hence, the angular momentum  becomes zero.

Thus, the choice of the point of the position vector can change the angular momentum