homework and exercises - Can angular momentum not be conserved in a straight line motion?

Consider a particle moving an a straight line, with constant velocity $v$. The angular momentum (pivot point $O$) can be calculated as

$$L=mr v_{\theta}$$ Where $v_{\theta}$ is the velocity perpendicular to the vector $r$ at each istant.

Now if I calculate the angular momentum in $A$ it get $L_A=mr_A v_x$, while in $B$ I get $L_B=mr_B v_y$.

In general $L_A\neq L_B$ but how can that be? How can angular momentum not be conserved? There are no forces, or torques!

I'm probably missing something big but I cannot see the mistake

Answer

If you draw similar triangles, then you'll find that $r_A/r_B = v_y/v_x$, and so the product $r_A v_x$ is equal to $r_B v_y$. Try drawing a line from the tip of your lower $\vec{v}$ vector to the tip of your lower $v_y$ component to see this.