Differences between vectors are important, e.g., in the definition of velocity where one has to take the difference between the position vectors of a particle, and divide this difference by a time elapsed.

This session will be more than just a 'glimpse' of differences between vectors. It could serve as a complete explanation of this topic. Nevertheless, in addition to reading the material here, you should also work through Vectors/Subtraction/Explain It where the process of vector subtraction will be explained with interactive simulations.

The process of subtracting vectors and the concept of the difference between two vectors are patterned on what is done with numbers. It may be helpful to review the situation with numbers before going on to vectors.

Suppose you want to find a number x such that

4 + x = 10.

Such a number exists, and it is called the difference between 10 and 4. We all know this difference is the number 6, but here we are interested in the formal way of writing this difference.

To arrive at this, let us solve the equation for x. To do so, we need to add the number (-4) to both sides of the equation. (-4) is defined as the number that added to 4 gives 0: (-4) + 4 = 0. Thus,

4 + x + (-4) = x = 10 + (-4).

Thus, the difference of 10 and 4 is equal to 10 + (-4). It is customary to abbreviate this by 10 - 4. The notation 10 - 4 is just a shorthand for 10 + (-4).

Thus, the difference of 10 and 4 is a number denoted 10 - 4 which

1. is the number that added to 4 gives 10,
2. is equal to 10 + (-4).

These two properties of the difference of two numbers also apply to the difference of two vectors.