Forces acting on a object can be approximated as vectors with a magnitude and direction. When there are unknowns in the system, the equations of equilibrium can be used to find those unknowns.
Recall in the previous section, the equation for equilibrium is
ΣF = 0
This equation can be expanded for each direction (only x- and y-direction is examined in this section, the next section includes the z-direction)
ΣFxi + ΣFyj
For the expanded equation, each direction must be in equilibrium, or
This means that when an object is in equilibrium, the components of forces acting along the arbitrary coordinate system will cancel each other out. These relationships can be used to determine the unknown forces using simple algebra.