# Vector subtraction

The emergency exit is being indicated by a horizontal vector heading to the right

In the study of Algebra we learned that the subtraction of two real numbers, for example x and y, can be given as follows:

*x – y = x + (-y)*

Where ** –y** is the

**opposite**of

**. In this way, for example:**

*y**7 – 3 = 7 + (-3) = 4*

We define the subtraction of two vectors in a completely similar way, starting from the concept of **opposite** . Let’s consider a non-zero (non-zero) vector . The opposite of is a vector that has the same magnitude and the same direction, but has a direction opposite to the direction of the vector .

We indicate the opposite of by ^{__} . In the figure above we have an example of a case where | | **= 3** . Therefore, | ^{__} | **= 3** and ^{__} have the opposite sense to . The opposite of the null vector is itself: .

Given then two vectors and , the difference between these two vectors is represented as follows:

= –

And it can be defined by:

= + (- )

Let us see, for example, the case of the figure below and determine the vector such that

= –

Therefore, we have:

= – = + (- )

In the figure above, we can see that the difference was obtained by adding with ^{__} . However, it is easy to see that the vector could be obtained by connecting the ends of and as in case 3 of the figure above, with direction from B to A.

The addition and subtraction of vectors has been defined in such a way that we can work with vector equations in a similar way as we did with equations between real numbers, passing a term from one side to another, changing its sign.