The vectors they are, on the ground of the physical , magnitudes defined by its point of application, its meaning, its direction and its value. Depending on the context in which they appear and their characteristics, they are classified differently.
The idea of unit vector refers to the vector whose module is equal to 1 . It should be remembered that the module It is the figure coinciding with the length when the vector is represented on a graph. The module, in this way, is a norm of mathematics that is applied to the vector that appears in a Euclidean space.
Another of the names by which the unit vector is known is normalized vector , and appears very frequently in problems of various fields, from mathematics to computer programming. It is possible to get the Internal product or Scalar product of two unit vectors finding out the cosine of the angle that forms between them. He product of a unit vector by a unit vector, thus, is the scalar projection of one of the vectors on the direction established by the other vector.
When you have a vector and you want to normalize it, what you do is look for a unit vector that has the same meaning and the same address Than the vector in question. The normalization of the vector is carried out by dividing the vector by its module. The result is a unit vector with the same direction and identical direction.
But what does it mean to divide the vector by its module? Let's not forget that the vector is defined by means of components, as many as dimensions there is in the space in which it is located. If we take a two-dimensional vector, expressed in the axes X and AND , then it will have a value for each one of them, such as being (4.3). It should be mentioned that these components are also known as vector terms.
Therefore, if we go back to the method to find the unit vector that consists of dividing the original by its module, simply we must take each of the components and divide them by that value , so that the final result offers us a module equal to 1. This may seem too abstract or arbitrary for people outside of mathematics, but once carefully analyzed it is absolutely logical. Let's see the explanation below.
If we rely on division rules for a moment, we will remember that every number is divisible by itself and by 1, and that if we divide it by itself the result we get is precisely 1. Now, in this case we are looking for a vector whose components orient it in the same direction of the original, but that generate a different length, more specifically, of value 1.
Returning to the procedure of dividing each component by the module, let's see how to reach that step in a logical way. First of all, it is necessary to remember that for calculate the module of a vector we rely on the Pythagoras theorem , since we consider the vector segment as the hypotenuse, and each of its components as the legs of the triangle.
Therefore, to calculate the module of the vector (4,3) we must obtain the square root of the sum of the squares of 4 and 3. This results in 5. To arrive at the unit vector, we must multiply everything by 1 / 5 (one fifth), so that next to the equality let's get 1 (the length of the normalized vector) and from the other we find 1/5 x (4.3) .
Finally, we can say that the components of the unit vector will be (4 / 5,3 / 5), and it is enough to apply the Pythagorean Theorem to verify that the module is in effect 1.
The use of unit vectors facilitates the specification of the different directions presented by the vector quantities in a given system of coordinates.