Moment of Force about a Point
The moment of a force is only defined with respect to a certain point P (it is said to be the "moment about P"), and in general when P is changed, the moment changes. However, the moment (torque) of a couple is independent of the reference point P: Any point will give the same moment. In other words, a torque vector, unlike any other moment vector, is a "free vector".
(This fact is called Varignon's Second Moment Theorem.)
The proof of this claim is as follows: Suppose there are a set of force vectors F1, F2, etc. that form a couple, with position vectors (about some origin P) r1, r2, etc., respectively. The moment about P is
M = r1 * F1 + r2* F2 + r3* F3 + ..
Now we pick a new reference point P' that differs from P by the vector r. The new moment is
M’ = (r1 + r)x F1 + (r2 + r)x F2 + (r3 + r)x F3 + ...
Now the distributive property of the cross product implies
M’ = (r1 x F1 + r2 x F2 + r3 x F3+ ….) + r x (F1 + F2 +F3 + ….)
However, the definition of a force couple means that
F1 + F2 +F3 +...= 0
M’ = r1 x F1 + r2 x F2 + r3 x F3+ .. =M
This proves that the moment is independent of reference point, which is proof that a couple is a free vector.