Mechanics

# 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 y . In this way, for example:

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.