Question

write the expression for the moment of force about a given axis

Answer 1

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.

Answer 2

Moment of force about a given axis= Force x perpendicular distance of force from the axis of rotation.