AA′ · cos α · AO = BB′ · cos β · OB,
(1)
and draw OC parallel to the vector-sum. Resolving AA′, BB′ each into two components parallel and perpendicular to OC, we see that the former components have a single resultant in OC, of amount
R = AA′ cos α + BB′ cos β,
(2)
whilst the latter components form a couple of moment
G = AA′ · AB · sin α = BB′ · AB · sin β.
(3)
Conversely it is seen that any wrench can be replaced in an infinite number of ways by two forces, and that the line of action of one of these may be chosen quite arbitrarily. Also, we find from (2) and (3) that
G · R = AA′ · BB′ · AB · sin (α + β).