I will now explain to you what the vectors are and the concepts related to the vectors you will need to calculate and operate with them, such as the components of a vector, the module of a vector, its direction and sense.
We will also look at the types of vectors that exist and with which type of vectors it is convenient to operate in mathematics.
Índice de Contenidos
- 1 What are vectors?
- 2 Components of a vector
- 3 Module of a vector
- 4 Direction and direction of a vector
- 5 Types of vectors
What are vectors?
Vectors are oriented segments, which start at the point that corresponds to the origin of the vector and end at another point, which is the end of the vector:
They are defined by two components: component x and component y, also called vector coordinates. They are two-dimensional magnitudes.
To express them analytically, they are usually expressed by a lowercase letter with an arrow on top, with its two components in parentheses (like coordinates):
The letters u, v, w and z are usually used, but are not mandatory and can be used in any letter.
Here are some examples of vectors:
This can also be expressed by two capital letters with an arrow on top, where the first letter is the point of origin and the second letter is the point where the end is located:
Thus, the vector AB will have its origin in point A and its end in point B:
In addition to the vector components, other characteristics that define the vectors are the module, the direction and the direction.
We’re going to see every one of them.
Components of a vector
What are the components of a vector?
These are the elements that define a vector, since knowing its coordinates, we know everything about it: module (which will have to be calculated), direction and sense.
Talking about vector components and vector coordinates is the same thing.
How to calculate the components of a vector?
To calculate the components of a vector, we need to know beforehand the coordinates of its origin and the coordinates of its end, since they will be calculated from them.
If the coordinates of the point of origin of a vector are:
And the coordinates of a vector’s end point are:
We calculate the coordinates of the vector, subtracting the coordinates of the end minus the coordinates of the origin:
To calculate the x component of the vector, we subtract the x coordinate of the end minus the x coordinate of the origin. In the same way, to calculate the component “y” of the vector, we subtract the “y” coordinate of the end minus the “y” coordinate of the origin.
Remember, always extreme less origin and each component is subtracted with its own: the x with the x and the y with the 0 and.
Let’s see an example:
We have the point of origin:
And the point corresponding to the end of the vector:
The coordinates of vector AB will correspond to the x-coordinates of the end minus the x-coordinate of the origin on the one hand and the y-coordinate of the end minus the y-coordinate of the origin on the other hand:
Now we operate within each component and we have it:
Module of a vector
What is a vector module?
The module of a vector is the distance from the origin to the end, so it corresponds to the length of the vector.
The letter of the vector (or letters) is represented enclosed between two bars:
The module of a vector, being a length, is always positive.
How is the module calculated?
The module of a vector is calculated from the coordinates of the vector using this formula:
The module of a vector is the square root of the squared x-coordinate plus the squared y-coordinate.
Does that formula ring a bell?
It is the same as for calculating the hypotenuse of a right triangle, since to obtain it, the Pythagorean theorem is applied in the triangle formed between the vector and its components, where the vector would be the hypotenuse:
You don’t need to learn where you get the formula from, but it’s important to know.
Let’s see an example of how to calculate a vector module. We have the next vector:
What is your module?
We apply the formula and it fits:
Now we operate and solve, getting the value of the module:
We leave it in the form of a root, since the root of 13 is not exact.
How to calculate the module of a vector with the coordinates of its origin and its endpoint
As we have seen before, the components of a vector are calculated from the coordinates of the points of its origin and end.
Therefore, we can calculate the module of a vector directly if we know the coordinates of the points of its origin and of its end, just by replacing in the previous calculation formula of the module, the components x and “y” by the subtraction of the coordinates of the end minus the coordinates of the origin:
Let’s look at it with an example.
We have a vector whose origin is in point A and its end is in point B:
We are going to calculate the module of the vector AB, which will be the root of the square of the subtraction of the coordinates x, minus the square of the subtraction of the coordinates “y”, always subtracting extreme minus origin:
Be very careful with the negative coordinates. We eliminate the parenthesis of -4 in the second term of the interior of the root:
We now operate within each parenthesis:
We square up and solve:
It is also possible to do the operation in reverse, i.e., knowing the vector module, calculate its components.
Direction and direction of a vector
As we have said before, a vector is a segment oriented, so the direction of a vector is the line to which that vector belongs.
All the parallel vectors, they all have the same direction.
The direction of a vector is where that vector is going, where the arrow is pointing.
The direction of vector AB will go from point A to point B, while the direction of vector BA will go from point B to point A:
Both vectors have the same direction, but opposite directions
Just so we’re clear, let’s compare the vector to a straight road. The road itself would be the direction of the vector and within the road, you can go in both directions, which would correspond to the directions of the vector.
Types of vectors
Vectors can be fixed, when their point of origin and end cannot move and free when their point of origin can be anywhere in space.
To define a fixed vector, you need to know the coordinates of its origin and the coordinates of its end. As it is fixed, it cannot be moved in any way.
Two parallel vectors, of the same module, but from different points of origin, are different vectors.
Vectors that have the same module, direction and direction, but have their origin at different points are called equipollent vectors.
The free vectors are those vectors whose origin can be located at any point and are those with which we work in mathematics, since they have a characteristic that makes them special.
And what makes a free vector so special?
Because we can move the origin of a free vector to the point we want, we can always assume that its origin is in the coordinate origin (0.0) and therefore the vector will be defined only by knowing the coordinates of the end.
You no longer need to subtract the coordinates of the origin from the coordinates of the endpoint to calculate the components of the vector.
So, when we say that the vector:
We are trying to say that its origin is (0.0) and its extreme (1.-4)