College Physics 3.8 – Order of addition of 3 vectors does not affect their sum

Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors A, B, and C, all having different lengths and directions. Find the sum A + B + C then find their sum when added in a different order and show the result is the same. (There are five other orders in which A, B, and C can be added; choose only one.)


Consider the three vectors shown in the figure below:

Three vectors are given: Vector A is directed to the right, vector B is directed upward, and vector C is directed to the left.
Figure 3.8A: The 3 given vectors

First, we shall add them A+B+C. Using the head-tail or graphical method of vector addition, we have the figure shown below.

The 3 vectors A, B, and C are added and the resultant force is also shown in red color directed upward.
Figure 3.8B: The resultant force of A+B+C

Now, let us try to find the sum of the three vectors by reordering vectors A, B, and C. Let us try to find the sum of C+B+A in that order. The result is shown below.

The sum of vectors A, B and C is shown with their resultant in red.
Figure 3.8C: The resultant of 3 vectors added in different order.

We can see that the resultant is the same directed from the origin upward. This proves that the resultant must be the same even if the vectors are added in different order.