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

Question

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 $$ ||p||*||r||*\sin{\theta} $$

Answer 1

Direction
As for the direction: in three dimensions any other axis would have a projection on the rotation plane, which would pose a problem, since for rotational motion all the directions in this plane should be equivalent. In my opinion, this is largely a matter of convention, although such a definition has lots of mathematical conveniences (e.g., we can add angular momenta).

Magnitude
The magnitude has to do with the part of the force that causes the rotation: it is the projection on the direction perpendicular to the radius, whereas the projection along the radius pulls on the axis.