In statics the resultant of a system of forces acting at various points on a rigid body or system of particles is a single force, acting at a single point, if one exists, which is equivalent to the given system.
Suppose that forces Fi act at points ri. The resultant would be a single force G acting at a point s. The systems are equivalent if they have the same net force and the same net moment about any point.
These condition are equivalent to requiring that
If , there is no net moment and the conditions are satisfied by taking and s=0.
If , the second condition is soluble only if is perpendicular to . Suppose that this necessary condition is satisfied. It is then the case that an appropriate s can be found.
We conclude that a necessary and sufficient condition for the system of forces to have a resultant is that