Find Torque about an Axis.

Explanation

(i) Consider a rigid body capable of rotationg about an axis AB as shown in Figure. Let the force F act at a point P on the rigid body.

(ii) The force F may not be on the plane ABP. The origin O at any random point on the axis AB is taken.
(iii) The torque of the force `vec(F)` about O is, `vec(tau) = vec(r) xx vec(F)`. The component of the torque along the axis is the torque of `vec(F)` about the axis. To find it, we should first find the vector `vec(tau) = vec(r) xx vec(F)` and then find the angle `varphi` between `tau` and AB. (Remember here, `vec(F)` is not on the plane ABP). The torque about AB is the parallel component of the torque along AB, which is `|vec(r) xx vec(F)| cos phi`. And the torque perpendicular to the axis AB is `|vec(r) xx vec(F)| sin phi`.