Final answer:
By adding three vectors A, B, and C in the order A + B + C and then B + C + A, we find that the resulting sums are identical, demonstrating that the order of addition of vectors does not affect their sum, which is a verification of the commutative property of vector addition.
Step-by-step explanation:
To demonstrate that the order of addition of three vectors does not affect their sum, we need to perform the operation with different orders and compare the results. Let's choose three vectors A, B, and C, with distinct lengths and directions. First, find the sum in the order A + B + C. Suppose vector A has coordinates (1,2), B is (3,4), and C is (5,6). The sum A + B + C is (1+3+5, 2+4+6) = (9,12).
Next, choose a different order to add these vectors, for example, B + C + A. Using the same vectors as before, the sum B + C + A is (3+5+1, 4+6+2) = (9,12), which is identical to the sum for A + B + C. This example shows that regardless of the order in which vectors are added, the resultant sum remains the same, confirming the commutative property of vector addition.