The unit vector is also known as a normalized vector.
Vectors are, in the field of physics , magnitudes defined by their point of application, their sense, their direction and their value. Depending on the context in which they appear and their characteristics, they are classified differently.
The idea of a unit vector refers to the vector whose module is equal to 1 . It should be remembered that the module is the figure that coincides with the length when the vector is represented in a graph. The module, in this way, is a mathematical norm that applies to the vector that appears in a Euclidean space.
Another name by which the unit vector is known is normalized vector , and it appears very frequently in problems in various areas, from mathematics to computer programming. It is possible to obtain the inner product or scalar product of two unit vectors by finding the cosine of the angle formed between them. The product of a unit vector and a unit vector, thus, is the scalar projection of one of the vectors onto the direction established by the other vector.
Normalization of a 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 sense and the same direction as the vector in question. Vector normalization is carried out by dividing the vector by its module. The result is a unit vector with identical direction and identical sense.
But what does it mean to divide the vector by its module? Let us not forget that the vector is defined by means of components, as many as there are dimensions in the space in which it is located. If we take a two-dimensional vector, expressed in the X and Y axes, then it will have a value for each of them, such as (4,3). It is worth mentioning that these components are also known as vector terms .
Therefore, if we return to the method to find the unit vector which consists of dividing the original by its module, we will simply have to take each of the components and divide them by said value , so that the final result offers us a module equal to 1 This may seem too abstract or arbitrary to people outside of mathematics, but once analyzed carefully it is absolutely logical. Let's see the explanation below.
The idea of a unit vector is used in mathematics and physics.
Operating with a unit vector
If we rely on the rules of division 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 obtain is precisely 1. Now, in this case we are looking for a vector whose components orient it in the same direction as the original, but that generate a different length, more specifically, a value of 1.
Returning to the procedure of dividing each component by the module, let's see how to get to that step logically. First of all, it is necessary to remember that to calculate the module of a vector we rely on the Pythagorean Theorem , since we consider the segment of the vector as the hypotenuse, and each of its components as the legs of the triangle.
Therefore, to calculate the magnitude of the vector (4,3) we must obtain the square root of the sum of the squares of 4 and 3. This gives us 5. To get to the unit vector, we must multiply everything by 1 /5 (one fifth), so that on one side of the equality we obtain 1 (the length of the normalized vector) and on 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 indeed 1.
The use of unit vectors facilitates the specification of the different directions that vector quantities present in a given coordinate system .