Now I will explain how to perform operations with free vectors. I will teach you how to add vectors, how to subtract vectors and how to multiply a vector by a number, both analytically and graphically.
In addition, we will see exercises solved in each of the cases.
Índice de Contenidos
How are two vectors added together?
To add two vectors, the x-coordinates on one side and the y-coordinates on the other side are added together.
So, if we have the vectors:
The sum of the vectors will be:
Let’s see an example: Add the following vectors u and v:
We add the x-coordinate of the vector v to the x-coordinate of the vector u and also the y-coordinate of the vector v to the y-coordinate of the vector u:
And so, as a resultant vector:
Let’s now see how to add the vectors graphically.
Vector Graphic Summation
The graphical addition of vectors can be done in two ways:
Let’s go with the first form:
1 – We have the vectors u and v:
We want to add v+u graphically. Therefore, we place the origin of u at the end of v:
We join the origin of v with the end of u and obtain the resulting vector v+u:
Try to do the graphical addition of u+v yourself and you will see that the result is the same ;).
Let’s go with the second way to add vectors graphically:
2 – We have the vectors u and v:
We place the two vectors in the same source:
A parallelogram is formed with these two vectors by drawing a line parallel to the vector u at the end of the vector vector and a line parallel to the vector vector u at the end of the vector u, as follows:
The union of the origin of both vectors with the intersection of the lines we have just drawn will be the vector sum u+v:
We now turn to the subtraction of vectors.
The subtraction of vectors is done in the same way as the sum of vectors.
To subtract two vectors, subtract the x-coordinates on one side and the y-coordinates on the other.
If we have the vectors:
The subtraction of the vectors v-u will be:
Let’s look at it with an example: Subtract the subtraction v-u, where v and u are the following vectors:
To find the subtraction of the vectors v-u we subtract on the one hand, from the x coordinate of v the x coordinate of u and on the other hand, from the y coordinate of v we subtract the y coordinate of u:
We operate within each coordinate, taking great care with the signs and the resulting vector v-u remains:
Vector subtraction can also be done graphically. I’ll explain it to you in the next section.
Vector Graphic Regression
As with the graphical addition of vectors, the graphical subtraction of vectors can be done in two ways. You will see that it is very similar to the sum but taking into account a very important detail.
Be the vectors v and u following:
Since we want to subtract v-u, the first step is to change the direction of the vector u:
Now we follow the same procedure as in the graphical addition of vectors, with the difference that the direction of the vector u is contrary to its original direction. It is the same as adding (-u).
Place the origin of the vector u with the opposite direction at the end of the vector vector:
We join the origin of the vector vector with the end of the vector u with the opposite direction and we obtain the resulting vector v-u:
We continue with the second way.
We have vectors v and u:
As before, as we want to subtract v minus u (v-u), we change the vector u’s direction:
Now we place the vector vector and the vector u with the opposite direction in the same origin:
We form a parallelogram, with these two vectors and drawing a line parallel to the new vector u, at the end of the vector vector and a line parallel to the vector vector v at the end of this vector u, as follows:
The union of the origin of both vectors with the intersection of the drawn lines will be the vector resulting from subtracting u-v:
Both with one form and the other, keep in mind that you must change the direction of the vector you want to subtract (never forget this) and then the procedure is the same as with the addition.
Product of a vector by a number
To multiply a vector by a number, multiply that number by each of the coordinates of the vector.
Be the vector:
And we want to multiply it by a number (which belongs to the set of real numbers):
The multiplication of the number by the vector is represented like this:
And multiply the number by each of the coordinates of the vector:
It’s the same as when you multiply a number by a polynomial.
Let’s look at it with an example. We have the next vector:
And we want to multiply it by 3:
To multiply the vector by 3, we represent it first:
Multiply 3 by each of the vector’s coordinates and operate within each coordinate to get the resulting vector:
When the vector to be multiplied is the null vector:
Then, the multiplication of any number with the vector null will be zero:
If the vector by which the number is multiplied is different from the null vector, that is, any other vector than the null vector:
So, if the number that multiplies the vector is zero, then the product of the number by the vector will be zero:
If the number that multiplies the vector is greater than zero:
The vector resulting from the multiplication of the number by the vector will be a vector with the same direction and sense as the vector, but its size will be as much larger than the value of the number.
For example, if we have the vector vector and multiply it by 3, the resulting vector will be 3 times larger:
Besides, the module of the vector resulting from the product of a number by a vector is equal to that number by the module of the vector:
Conversely, if the number by which the vector is multiplied is less than zero:
The resulting vector will be a vector with the same direction but in the opposite direction.
In this case, the vector module resulting from the multiplication of a number by a vector is equal to the absolute value of the number by the vector module:
Be very careful not to confuse the module with the absolute value, since both cases are represented the same (enclosing the element between two bars).