The Resolution of a Force into a Force and a Couple

The action of a single force can always be duplicated by an equal parallel force plus a couple. Suppose that it is desired to resolve the force $𝐅$ of Fig. 1a, whose point of action is $B$, into a parallel force $𝐅$ passing through the point $A$, plus a couple. We introduce into the system the two equal and opposite forces $𝐅$ and $-𝐅$ acting at the point $A$. Since these two forces form a system in equilibrium they can be added without altering the motional effects of the original $𝐅$ upon the system. The circled forces in Fig. 1b can now be treated as a couple $𝐂=𝐚\times 𝐅$, and the original single force has been resolved into an equal parallel force and a couple.