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The above provides either R =0, or R is perpendicular to both the directions. If R is perpendicular to both the directions, then either the directions make 00 or 1800 with each other, which is not the case. Hence, R =0. To illustrate this one example is presented.
Fig 1: A body in an inclined plane
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Equation of equilibrium

Proof of sufficiency condition:

Assume that the resultant force makes an angle of and with x and y -axis respectively. The conditions of equation (2.4) will imply that