LESSON 3: 3D Sum Of Forces


While it is possible to sum the forces in a three dimensional plane by taking the component parts (3 component parts, x, y, and z directions) of each vector and simply summing the forces as you would in a two dimensional plane, this process can be somewhat tedious and inefficient. It is more difficult to break three dimensional vectors into their component parts than two dimensional vectors. So, typically in statics we use an alternate method to analyze three dimensional vectors. This method is essentially the same thing as simply breaking the vectors into component parts and then summing them, but it is generally recognized as an easier to accomplish method.

Let’s say you are given a three dimensional problem where there are three ropes with unknown tensions holding up a weight (this is a standard type of problem). The first step in this alternate method is to
find the unit vector of each rope in the three dimensional plane. You can usually find the unit vector by analyzing the lengths of each rope. The next step is to multiply the unknown force in each rope by its unit vector (create a variable for each unknown force). You should then group the x-components of each unit vector together, the y-components of each unit vector together, and the z-components of each unit vector together to form three equations. With these three equations you should be able to solve for the three variables of unknown force in the rope.

The graph to the right shows three vectors (vectors A, B, and C), whose lengths are given but forces are unknown, holding up a 46 N force (purple vector). To find the force in each vector first find the length of each vector’s component parts. For example, vector A goes 2 m in the x-direction, 1 m in the negative y-direction, and 6 m in the z-direction, so the length vector for vector A looks like this: 2 i - 1 j + 6 k . Next, find each vector’s unit vector:



After finding each unit vector, then multiply each unit vector by its respective vector’s unknown force:





Now we can set up three equations with three unknown variables. Group the x-components together, y-components together, and z-components together to set up the three equations:

You can then solve for the unknown variables. When I solved for the unknown variables using my calculator I found that A= 6 N , B= 3.5 N , and C= .5 N If you would like to view more examples of how to work this type of problem, two example problems are included here: click here for example 1 and click here for example 2