newtonian mechanics - How to choose origin in rotational problems to calculate torque?

We know that $\text{Torque} = r \times F$ and $r$ is the position vector. But the position vector depends on the choice of the coordinate system and in turn on the choice of origin. So, where should we take the origin?

Also, do torque, angular velocity and angular acceleration point of out the plane of rotation for 2D objects because otherwise they wouldn't have constant direction?

Many sources (including my textbook) seem to say that the origin should lie on the axis and that it wouldn't make a difference where it is on the axis.. but I don't get why it shouldn't, since position vector would be different from different origins and so the torque, according to me, might come out to be different.

Answer

In order to calculate the torque, $\vec\tau=\vec r\times\vec F$, one can choose any origin $O$. The torque then is said to be calculated with respect to $O$ and it is dependent of this choice. In particular, if the sum of all external forces on the system vanishes then the resultant torque is independent of $O$.

For the second question, note that when a particle is rotating in a fixed plane, say $xy$ plane, and the forces acting on it are also in this plane then the torque is in the $z$ direction because a vector product with force must be orthogonal to it. Similarly the expressions for the angular velocity and angular acceleration also satisfy vector product relations, $\vec v=\vec\omega\times\vec r$ and $\vec a=\vec\alpha\times\vec r+\vec\omega\times\vec v$. As you can see, angular velocity has to be perpendicular to the velocity and angular acceleration has to be perpendicular to the acceleration.