Vectors in mathematics. Representation and operations

Representation of vectors in mathematics

Mathematically, a vector is represented by a letter with an arrow on top. Graphically, it is represented by an arrow in n-dimensional space, where each component of the vector represents a quantity in a specific direction. Thus, a vector is defined as a sequence of ordered numbers. For example, a two-dimensional vector can be represented as (x, y), where “x” is the component on the x-axis and “y” is the component on the y-axis. In a three-dimensional space, the vector is represented as (x, y, z), where “z” is the component on the z-axis.

Operations with vectors

Different operations can be performed with vectors, such as addition, subtraction, multiplication by a scalar, scalar product and vector productOperations with vectors allow you to manipulate and combine these mathematical entities to solve numerous problems in mathematics, physics, engineering and other sciences. The most important operations that can be performed with vectors are explained below:

Vector addition

The addition of two or more vectors consists of obtaining a new vector that results from placing the vectors one after the other, respecting their directions and senses. The resulting vector goes from the origin of the first one to the end of the last one. Algebraically, the addition is performed by adding the corresponding components of each vector. For example, if you have vectors a = (a₁, a₂) and b = (b₁, b₂), their sum will be a + b = (a₁ + b₁, a₂ + b₂). This operation is fundamental for calculating total displacements, net forces, and many other composite quantities.

Subtraction of vectors

Subtraction of vectors is interpreted as the sum of the first with the opposite of the second. The opposite of a vector has the same magnitude but opposite direction. If a and b are vectors, then a – b = a + (-b). In components, this is equivalent to subtracting each component: (a₁ – b₁, a₂ – b₂).

Multiplication by a scalar

Multiplying a vector by a real number (scalar) changes its magnitude but not its direction (except if the scalar is negative, in which case the direction is reversed). For example, if k is a real number and a is a vector, then ka = (k-a₁, k-a₂). This is useful for resizing forces, velocities, or any vector quantity.

Scalar product

Also called dot product, it is an operation that associates two vectors with a real number. It is calculated by multiplying the corresponding components and adding the results: a – b = a₁-b₁ + a₂-b₂ (in two dimensions). The scalar product measures the degree of alignment between two vectors and is fundamental in work calculations and projections.

Vector product

It is only defined in three dimensions and results in another vector perpendicular to the two original vectors. It is widely used in physics to calculate moments, forces and oriented areas. If a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the vector product is a vector whose components are calculated with the determinant of a matrix.

These operations make it possible to model and solve complex situations involving several vector quantities. Learning to operate with vectors facilitates data analysis and problem solving in multidimensional contexts.

Vector applications

Vectors are widely used in physics, engineering and many other scientific disciplines. They are used to represent physical quantities with direction and modulus, such as forces, velocities and displacements. They are also useful in image processing, artificial intelligence and machine learning, where they are used to represent characteristics or attributes of data. Their study and understanding make it possible to model physical phenomena and solve problems in various fields, which makes them a fundamental tool in the analysis and representation of data and phenomena in the real world.

The online vector simulations on this page will be extremely useful to improve your understanding of this important part of mathematics.