A couple is defined as a pair of forces having equal magnitude and opposite direction with different but parallel lines of action. Since the total force in any direction is zero, the net effect of a couple is a pure moment, or torque. The magnitude of this moment is the magnitude of one of the forces times the perpendicular distance between the forces,

     M = F d

The moment vector is perpendicular to the plane containing the lines of action of the two forces.

Using a vector approach, the moment of a couple is defined as

     M = r × F

Here r is the position vector from one force to the other, and F is the force to which r is directed.

Two systems are equivalent if and only if the sum of forces for the two systems is equal and the sum of moments about an arbitrary point in both systems is equal:

     (ΣF)₁ = (ΣF)₂

     (ΣM_p)₁ = (ΣM_p)₂  

Using the concept of equivalent systems, any system of forces and moments can be replaced by a single force and couple at an arbitrary point P. Applying the conditions for equivalent systems gives

     F_R = ΣF                                                  (1)

     M_RP = ΣM_P                                             (2)

Here each M_P is given by the equation

     M_P = r_PF × F                                           (3)

Single Force Resultant

If the force and couple of an equivalent force-couple system are perpendicular to one another, we have the equation

     F_R • M_RP = 0

The system can be reduced to a single force F_R located at a point Q. The moments of the two systems can be equated about the point P giving,

     M_RP = r_PQ × F_R                                      (4)

By substituting Eqs. 1, 2, and 3 into Eq. 4; performing the cross products; and equating the i, j, and k components; the three scalar equation are,

     Σ(r_PFy F_z - r_PFz F_y ) = r_PQy ΣF_z - r_PQz ΣF_y

     Σ(r_PFz F_x - r_PFx F_z ) = r_PQz ΣF_x - r_PQx ΣF_z

     Σ(r_PFx F_y - r_PFy F_x ) = r_PQx ΣF_y - r_PQy ΣF_x

Solving this system of three equations for the three unknowns r_PQx, r_PQy, and r_PQz gives the position vector r_PQ of the force F_R,

     r_PQ = r_PQxi + r_PQyj + r_PQzk

     F_R = ΣF