In the 2D equilibrium section, it was shown that the sum of all forces and the sum of all moments acting on a rigid body in equilibrium must be equal to zero,

     ΣF = 0              ΣM = 0

Using rectangular coordinates, these vector equations can be expanded into

     ΣF = ΣF_xi + ΣF_yj + ΣF_zk

     ΣM = ΣM_xi + ΣM_yj + ΣM_zk

These also represent six independent scalar equations,

     ΣF_x = 0      ΣF_y = 0     ΣF_z = 0

     ΣM_x = 0      ΣM_y = 0     ΣM_z = 0

Note that the sum of the moments can be evaluated about any point. Further equations can be obtained by summing the moments about other points, but they will not be independent of the first moment equations.

Because there are not more than six independent scalar equations, it is not possible to solve for more than six unknowns.