# Why is the angular momentum defined as the cross product of linear momentum and the radius?

I first searched why is Torque defined as a cross product and a lot of people answered that it is defined as a cross product because angular momentum is defined that way and it is only logical that Torque and angular momentum have the same direction.

I know the cross product is used for the direction of angular momentum but why does the direction of angular momentum have to be in the direction of axis of rotation and even if it has to be in the direction of axis of rotation then why does it have to be a cross product and not some other mathematical product.

also why is the magnitude of angular momentum defined as $$||p||*||r||*\sin{\theta}$$

• The last point about the magnitude just comes from the definition of the cross product. – Paul G. Mar 30 '20 at 11:51