ES-115 Three Force Equilibrium
If three forces are applied to a body they must be coplanar and concurrent or parallel for equilibrium to be possible.
Let us apply three forces to a body:

Since two lines define a plane, let us let and define the x-y plane. is then the only force with a z component.

For equilibrium...

Since the third force has no z component the forces must be coplanar. Let us assume that intersect at 0. If we take moments about 0...

Since is not equal to zero, d must be zero or parallel to .
If it is zero, the three forces are concurrent, meaning they all pass though the same point.

If and are parallel, they define an axis, let us say the x-axis. then, is the only force with a y component, hence the equation shown to the left...

Therefore the three forces are parallel.

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