Hi Johann,

Is your problem with torque as a vector the fact that if the torque points sideways out a bicycle wheel, that the wheel just rotates instead of moving sideways?

If so, then the solution is that a vector is a mathematical construct. It has a magnitude and direction and those two things allow it to model a variety of physical effects. When modeling a force, which is a push or a pull, it points in the direction of the push or pull, so you expect the object to move in that direction.

In the case of a rotation, how can you represent that rotation? You can tell me in words that the object is rotating either clockwise or counterclockwise, and you can tell me what axis it's rotating about. Mathematically, that can be represented by drawing a line segment along the axis of rotation. Then to designate clockwise or counterclockwise, you can put the arrowhead on one end or the other of that segment. What you've drawn is a vector, but it represents rotation about the axis, not a push or pull along the axis. That's why torque is a vector: it's a nice way to represent clockwise or counterclockwise rotation about an axis.

(To be technical, torque is a pseudovector. This is a technical term for how it's magnitude and direction change as you change coordinate systems. It doesn't have a lot to do with answering this question, but I bring it up for completeness.)