Vectors are, in the field of physics, quantities defined by their point of application, direction, address and 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 modulus is the number that coincides with the length when the vector is plotted on a graph. The modulus, therefore, is a rule of mathematics 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 often in problems in various fields, from mathematics to computer programming. It is possible to obtain the inner product or dot product of two unit vectors by finding the cosine of the angle formed between them. The product of a unit vector by a unit vector, therefore, is the scalar projection of one of the vectors in 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 sense and the same direction as the vector in question. Vector normalization is performed by dividing the vector by its module. The result is a unit vector with the same direction and the same sense. But what does it mean to divide the vector by its modulus? 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 found. If we take a two-dimensional vector, expressed on the X and Y axes, it will have a value for each of them, like (4,3). It should be mentioned that these components are also known as vector terms. So, if we go back to the method to find the unit vector that consists of dividing the original by its modulus, we will only have to take each of the components and divide them by this value , so that the final result gives us a modulus equal to 1 This it may seem too abstract or arbitrary to non-mathematicians, but when you look closely, it makes perfect sense. Let’s see the explanation below. If we rely on the division rules for a moment, we will remember that every number is divisible by itself and by 1 , and that if we divide 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 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 reach this step logically. First of all, it is necessary to remember that to calculate the magnitude of a vector we are based 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 take the square root of the sum of the squares of 4 and 3. This gives us the result 5. To arrive at the unit vector, we must multiply everything by 1/5 (a fifth ), so that on one side of the equality we get 1 (the length of the normalized vector) and on the other side we get 1/5 x (4,3) . Finally, we can say that the components of the unit vector will be (4 / 5,3 / 5), just by applying the Pythagorean Theorem to verify if the modulus is in fact 1. The use of unit vectors facilitates the specification of the different directions that vector quantities present in a given coordinate system.