# Equilibrant/Examples/100kg at 150, 75kg at 60, 50kg at -45/Proof 2

## Example of Equilibrant

Three forces $\mathbf F_1, \mathbf F_2, \mathbf F_3$ act on a particle $B$ at a point $P$ embedded in the complex plane:

 $\displaystyle \mathbf F_1$ $=$ $\displaystyle \polar {100 \, \mathrm {kg}, 150 \degrees}$ $\displaystyle \mathbf F_2$ $=$ $\displaystyle \polar {75 \, \mathrm {kg}, 60 \degrees}$ $\displaystyle \mathbf F_3$ $=$ $\displaystyle \polar {50 \, \mathrm {kg}, -45 \degrees}$

The equilibrant $\mathbf E$ of $\mathbf F_1, \mathbf F_2, \mathbf F_3$ is:

$\mathbf E = \polar {80.8 \, \mathrm {kg}, -80.2 \degrees}$

## Proof

Using the Parallelogram Law to add $\mathbf F_1$ and $\mathbf F_2$ we arrive at $\mathbf F_1 + \mathbf F_2$.

Again using the Parallelogram Law to add $\mathbf F_1 + \mathbf F_2$ and $\mathbf F_3$ we arrive at $\mathbf F_1 + \mathbf F_2 + \mathbf F_3$.

The equilibrant $\mathbf E$ can then be found by taking the negative of $\mathbf F_1 + \mathbf F_2 + \mathbf F_3$.

$\blacksquare$