Example 11: In this example we will analyze a three-dimensional or non-coplanar force system. We want to determine the tension in each cable to hold the 250-lb crate in equilibrium. Note that points B, C, and D are all in x-y plane.
Solution: We begin the analysis by drawing the free-body diagram of the ring at point A. The forces acting on the ring include the three cable forces as well as the weight of the crate.
Since the forces are in three dimensions, we use the corresponding direction cosines in formulating the equations of equilibrium in x,y,z directions. Knowing the coordinates of points A, B, C, and D, we calculate the length of each cable as
Having found the length of each cable, we can now write the equilibrium equations in terms of the individual force components. We have to be careful with the direction of each component.
We can either use the back substitution to solve for cable force, or simply write the three equations in matrix form and solve them simultaneously as shown below