Resultant of Coplanar Forces: When we are examining a system involving two or more forces, we are usually interested in finding the resultant force in terms of its magnitude as well as direction. The graphical, trigonometric, and vector approaches discussed earlier can be applied to problems involving coplanar (two-dimensional) forces. We will expand on this discussion with the help of the following examples.
Resultant of Non-Coplanar Forces: The discussion in this section applies mainly to systems involving more than two concurrent forces. A simple experiment of holding two pencils (as a model of two vectors) end to end and rotating them around at various angles will show that two concurrent vectors are always coplanar. Therefore, we could use the graphical approach in finding the resultant. The complexity arises, however, when the two forces are located in a plane other than xy, xz, or yz plane, or when the system involves three or more non-coplanar forces. In that case, it would be easier to use trigonometric or vector approach to find the resultant force. Here, we make use of direction cosines and/or unit vector to help define the exact direction of a force vector. The analysis of such a system is demonstrated in the following examples.