Operations with vectors. Exercises resolved step by step.

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.

Vector Sum

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:

operations with vectors

vector exercises

The sum of the vectors will be:

vectores ejercicios

Let’s see an example: Add the following vectors u and v:

resolved vector exercises

resolved vector exercises

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:

suma y subtraction de vectores ejercicios resueltos

And so, as a resultant vector:

suma de vectores ejercicios

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:

vectors resolved exercises

We want to add v+u graphically. Therefore, we place the origin of u at the end of v:

suma de vectores ejercicios resueltos

We join the origin of v with the end of u and obtain the resulting vector v+u:

vector exercises solved graphically

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:

operations with exercise vectors

We place the two vectors in the same source:

vector addition exercises

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:

exercises with vectors

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:

vector operations

We now turn to the subtraction of vectors.

Vector subtraction

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:

resolved vector addition exercises

vectores fisica ejercicios resueltos

The subtraction of the vectors v-u will be:

operations with vectors exercises solved

Let’s look at it with an example: Subtract the subtraction v-u, where v and u are the following vectors:

method of parallelogram solved exercises

exercises 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:

operations between vectors

We operate within each coordinate, taking great care with the signs and the resulting vector v-u remains:

operations with vectors examples

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.

First form:

Be the vectors v and u following:

sum of vectors analytical method exercises solved

Since we want to subtract v-u, the first step is to change the direction of the vector u:

module of the resulting vector exercises solved

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:

vector problems solved

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:

calculation of vectors exercises solved

We continue with the second way.

We have vectors v and u:

exercises of resolved physical vectors

As before, as we want to subtract v minus u (v-u), we change the vector u’s direction:

multiplication of vectors exercises solved

Now we place the vector vector and the vector u with the opposite direction in the same origin:

suma de vectores ejemplos

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:

multiplication of vectors examples solved

The union of the origin of both vectors with the intersection of the drawn lines will be the vector resulting from subtracting u-v:

solved vector problems

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:

suma de vectores metodo grafico ejercicios resueltos

And we want to multiply it by a number (which belongs to the set of real numbers):

vector problems

The multiplication of the number by the vector is represented like this:

exercises sum of vectors

And multiply the number by each of the coordinates of the vector:

resolute exercises of basic physical vectors

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:

operations with physical vectors

And we want to multiply it by 3:

examples of vector operations

To multiply the vector by 3, we represent it first:

vector operations exercises

Multiply 3 by each of the vector’s coordinates and operate within each coordinate to get the resulting vector:

how to solve physical vectors step by step

When the vector to be multiplied is the null vector:

subtraction of vectors resolved exercises

Then, the multiplication of any number with the vector null will be zero:

operations of vectors addition subtraction and multiplication

If the vector by which the number is multiplied is different from the null vector, that is, any other vector than the null vector:

exercises of addition and subtraction of vectors

So, if the number that multiplies the vector is zero, then the product of the number by the vector will be zero:

suma y subtraction de vectores ejercicios

If the number that multiplies the vector is greater than zero:

exercises of resolved physical vectors 1

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:

multiplicacion de vectores

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:

sum of vectors polygon method solved exercises

Conversely, if the number by which the vector is multiplied is less than zero:

exercises solved vectors

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:

examples of vector addition

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).