A Very Useful Remark About The Comparison Method

The construction of the difference - by the comparison method relates directly to the physical interpretation of the difference as a change. The points at the tips of the two vectors and can be interpreted as positions of a particle at two different times, the point at the tip of being the position at an earlier time and that at the tip of at a later time.

The two vectors and can then be interpreted as the position vectors of the particle at the two times, relative to a fixed origin at the point where the two tail ends join. Then -, the later minus the earlier position vector, represents the change in the particle's position vector during the given time interval, and this change points in the direction in which the particle's position is changing, which is from to . (Note that the change in a quantity is always the later value minus the earlier value. E.g., if a number changes from 4 to 10, it changes by 6, and this change is equal to 10 - 4, the later minus the earlier value.)

More generally, when you have to construct the difference - of two vectors, think of the first vector, , as the later value of some vector quantity and the second one, , as the earlier value, think of - as the change in the quantity, and then draw and tail-to-tail. The change in the quantity, i.e., - , will point from the tip of the earlier value of the quantity to the later value .

The add-the-negative method allows no such interpretation because the green difference vector in that construction does not point from the earlier to the later value of any quantity. It points from the tail end of vector to the tip of vector -, and these points do not have a physical interpretation.