Question

How is the direction of torque perpendicular to the plane of rotation (according to the cross product of F and r)? It seems very counterintuitive and doesn't seem to make sense.

Answer 1

Because that force is perpendicular to the direction towards the rotation-centre. Not to the turning direction. The bolt does indeed turn in the same way as the force pulls it.

When you define a torque vector direction, you have a problem. You can't define a vector direction as something that turns around. The direction must be along a straight line. So instead of choosing the torque "turn", we could choose the torque axis as the vector direction.

Have a look at this picture:

The axis is vertical through the bolt along the two upwards/downwards arrows. If you choose to define the torque vector direction along this axis, all fits. We just have to remember that choice.

Torque is:

τ⃗ =F⃗ ×r⃗

The force vector F⃗ times the vector towards the rotation-centre r⃗ gives the torque vector. The result of a cross-product is mathematically a vector pointing vertically upwards, so this fits perfectly to that choice. The torque vector τ⃗ that you get from this calculation has the torque magnitude but the torque-axis direction.

As long as you remember this choice - this definition - all is good. Everytime you hear "the direction of the torque is horizontal", you know that this is only the axis of the torque; the torque (the turn) is then upright.