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I have learnt in class about finding the moment of a force F about a point A using the formula

M = r × F

where r is the vector directed from A to the point of origin of F

A textbook problem requires me to find the moment of a force about an axis. I had a look at https://en.wikipedia.org/wiki/Bending_moment#Computing_the_moment_of_force and found another formula:

M = e · (r × F)

However, now I cannot understand what r to take in this formula. The problem in my textbook requires me to find the moment of a force of 276.85N directed from (8,-3,0) to (0,0,6) along an axis directed from (12,0,0) to (0,4,-6). The answer provided is 1000Nm.

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From the page you cite:

r is the position vector from the reference point (O) to the point of application of the force (A).

In other words, r is the vector joining a point of the axis of rotation, and the point where the force is applied.

The vector e is a unit vector of the rotation axis. Projecting on e, as done in your second formula, ensures you can take any reference point (O) on the rotation axis, and still obtain the same result.

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