### Vector Geometry

Many ways problem

## Problem

In each situation below, three forces are acting on a particle. Given two of the forces and the fact the particle is in equilibrium, find the third force.

$\mathbf{a} = \quantity{-3\mathbf{i} + 2\mathbf{j}}{N}$

$\mathbf{b} = \quantity{4\mathbf{i} + \mathbf{j}}{N}$

$\mathbf{a}$ has magnitude $\quantity{10}{N}$ and direction $45^{\circ}$ anticlockwise from the positive $x$ axis

$\mathbf{b}= \quantity{6\mathbf{i} - 3\mathbf{j}}{N}$

Is it possible to use a different approach for each situation?