Q.What is the torque of the force F = 3i - 2j + 4k acting at a point r = 2i + 3j + 5k about the origin ? (i, j, k are unit vectors)

The idea of torque about a single point extends naturally to torque about an axis of rotation, which is what actually matters for a rigid body constrained to spin about a fixed line (like a door on its hinges, or a wheel on its axle) rather than free to rotate about any arbitrary point.

Setup. For a rigid body free to rotate about a fixed axis $AB$, choose any point $O$ lying on that axis as the origin. A force $\vec F$ acting at some point $P$ on the body (not necessarily in the plane containing the axis and $P$) has torque $\vec\tau=\vec r\times\vec F$ about $O$, as usual. The torque about the axis $AB$ is defined as the component of this torque vector along the axis — found by computing the angle $\phi$ between $\vec\tau$ and $AB$, giving a parallel component $|\vec r\times\vec F|\cos\phi$ (which actually turns the body about the axis) and a perpendicular component $|\vec r\times\vec F|\sin\phi$ (which instead tends to tilt the axis itself, producing precession — a phenomenon visible in a spinning top as it slows down, but beyond the scope of this course).

Key simplifications used from here on, since the axis is assumed held fixed by external constraints that automatically cancel any perpendicular component:

1. Only forces lying in planes perpendicular to the axis (and not intersecting it) are considered.
2. Only position vectors perpendicular to the axis are considered.
3. A force parallel to the axis produces a torque perpendicular to the axis, and can be ignored for rotation about that axis. …