........................(2.4)

Condition is necessary: Let R make an angle

with the x-axis. The sum of the resolved components of the forces is equal to the resolved component of resultant. Thus,

............................(2.5)

.............................(2.6)

Condition is sufficient: Assume that Eq. (2.5) and Eq. (2.6) are satisfied. Squaring both and adding,

Hence, R will be zero.

It is not necessary that directions x and y should be orthogonal. Only the angle between them should not be 0⁰ or 180⁰ . Necessity condition is obvious, because the resolved component of a zero force in any direction is zero.

Equation of equilibrium

Equilibrium of Coplanar Concurrent forces:

If all the forces in a system lie in a single plane and pass through a single point, then the system constitutes a coplanar concurrent force system.

The resultant R of a system of forces is zero if and only if it's resolved components in two orthogonal directions x and y are zero. Moment about the common point of forces is obviously zero.