Torque as a Cross Product

A turning effect needs a direction, and the only direction that makes sense for something spinning is along the axle it spins about. A force applied off to the side of a pivot does not point along the axle itself, so physicists define torque as a new vector built from r (the position of the force's application point, measured from the pivot) and F (the force). The cross product r x F automatically produces a vector perpendicular to both r and F, which is exactly the axle direction.

The right-hand rule assigns the direction: point your fingers along r, curl them toward F, and your thumb points along tau. If r is along x and F is along y, their cross product points along z, out of or into the page. Magnitude follows the same pattern as any cross product: |tau| = |r||F|sin(theta), where theta is the angle between r and F. For r = 2i meters and F = 5j newtons, tau = r x F = 2i x 5j = 10k newton-meters, straight out of the page.

A force diagram for torque has to show more than arrows for direction and size. It has to mark exactly where each force contacts the object, because r is measured from the pivot to that contact point. Slide the same force to a different point of application and r changes, so the torque changes even though F does not. This is also why moving the pivot matters: tau = r x F is always computed about a chosen point, so picking a new pivot gives a new r for every force and can change every torque's value, sometimes even its sign. This sets up the next step, where multiple torques about one pivot get summed into a net torque for rotational Newton's second law.