Question

If three concurrent forces are in equilibrium, what is the relation between any one of the three forces and the resultant of the other two forces?

Answer 1

Let $\mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_3$ be three concurrent forces (bold denotes vectors). Since the body is in equilibrium, the following equation should hold:

$$\mathbf{F}_1+ \mathbf{F}_2+ \mathbf{F}_3 = 0$$

Let $\mathbf{F}_{23}$ also be the resultant of forces $\mathbf{F}_2$ and $\mathbf{F}_3$: $\mathbf{F}_{23} = \mathbf{F}_2+\mathbf{F}_3$. Then obtain:

$$\mathbf{F}_1 + \mathbf{F}_{23} = 0\\ \mathbf{F}_1 = -\mathbf{F}_{23}$$

Thus, one of the three forces should have equal magnitude and opposite direction with the resultant of the other two forces.

Answer. Equal mabnitudes and opposite directions.