How To Add And Subtract Vectors

We’ll show you some ways to add and subtract vectors. Remember that vectors are tools that allow us to represent vector magnitudes, that is, those that not only need a quantity but also a direction and a sense.

Geometrically they are represented by arrows. For example, if we name the magnitude ” displacement ” we know that we are referring to a movement, going from one place to another. But to where?, in what direction and in what sense? So that you understand better, imagine that you are standing on a corner and you are very clear about how the cardinal points are located and I want to show you how to get to a certain point close to you. It is not the same if I tell you: scroll15 steps and then 21, what if I tell you: move 15 steps north and then 21 steps east. In the first case you will be able to get anywhere because I have not given you the direction or direction of that displacement, while in the second case it is correctly indicated and you will arrive at the exact point.

Each displacement you made, first to the North and then to the East can be represented by a vector and if you imagine a new vector that starts at your starting point and ends at your arrival point, that is the vector “sum” of the two first. Now we will show you how to add and subtract vectors forgetting about the concrete example.

Instructions for Adding and Subtracting Vectors

1. Vectors are represented by arrows, have an origin, and an end. The elements of the vector are: the intensity or module (it is the measure of the vector), the direction (it is the straight line that contains it) and the direction (indicates which way and is given by the arrowhead).
2. Vectors can be added, subtracted, multiplied. The addition or subtraction of vectors results in another vector(called the resultant) which can be obtained through different procedures.
3. There are different methods to add and subtract vectors both in the plane and in space. There are geometric methods and numerical methods. We will see one of each and several examples.
4. If we want to calculate the resultant of the addition or subtraction of vectors, we can do it through its Cartesian coordinates. Remember that vectors can be given by their Cartesian coordinates(x ; y ), for example, the vector v = (4 ; 3) starts at the origin of coordinates and ends at the point 4 units to the right ( because the number is positive) and 3 up (because the number is positive). Let’s generalize this: the  x -coordinate indicates units to the right (+) or to the left (-) and the y-coordinate to up (+) or down (-).
5. We can add or subtract vectors given by their Cartesian coordinates, either in the (x,y) plane or in ( x; y; z) space.
6. To add two or more vectors we simply have to add all the x components on the one hand and all the y components on the other. If we were in space we would also add the z
7. In the plane, for example, we have the vectors u and v given by their coordinates and we want to add them:
8. u =(-5; 6) v = ( 7;-1) u + v= ( 2 ; 5 ) because : -5 + 7 = 2 and 6 +(-1) = 6 – 1 = 5
9. If the vectors are located in space: u =( -3 ; 4 ; 2) v =( 6 ;-9 ; 8) u + v= ( 3 ; -5 ; 10 ) because : -3 + 6 = 3, 4 – 9 = -5 and 2 + 8 = 10
10. We can combine addition and subtraction of vectors in the plane, for example:
11. u= ( 4; -6), v = ( 7; 9) and w = ( -1; 1), u – v + w   = ​​u + (-v) + w = ​​( -4 , -14)
12. Because: 4 – 7 – 1 = -4 and -6 – 9 + 1 = -14
13. We can also combine addition and subtraction of vectors in space, for example:
14. u = (-3 ; -4 ; 4) v =( 6; – 3 ; 8) w =( 9 ; -7 ; -2) u + v – w= ( -6 ; 0 ;14 ) because: -3 + 6 + (– 9) = -3 + 6 – 9 = -6 , -4 -3 + (+7) = -4 -3 +7 = 0 and 4 + 8 + (+2) = 4 + 8 + 2 = 14
15. If we want to add two vectors u + v geometrically, we must place one vector after the other, that is, translating them parallel to themselves, so as not to change their direction or sense, and making the origin of the second vector coincide with the end of the first.  Then the sum vector (u + v) will be a new vector whose origin is the origin of the first, in this case of u, and whose endpoint is the endpoint of the second, in this case of v. Thus we can add two, three or more vectors, always placing one after the other, for example:
16. To subtract vectors we will only have to add to the first the opposite of the second.  The opposite of a vector has the same measure but the opposite direction. Let’s see an example: ab = a+(-b)
17. We can also combine addition and subtraction, for example: if we want to solve: u + v – w we must add the vectors u, v and the opposite of w , that is (-w)

What do you need to add and subtract vectors?

• Smooth paper
• Pencil, eraser, colors
• Ruler and square to draw parallels.

Tips for Adding and Subtracting Vectors

• When you do the calculations of the coordinates, be careful with the signs of the numbers, remember that when we say that we add integers we are doing it with the sign that each one has.
• Reinforce the practice of drawing parallel lines using a ruler and square so as not to make mistakes, since to obtain correct results when you add or subtract vectors geometrically you need precision in drawing parallel lines.