The line in the Cartesian plane. Equations and characteristics

What is a line in the Cartesian plane

A line in the Cartesian plane is an infinite set of points that follow the same direction and have no curvature. In mathematics, it is represented by algebraic equations and is one of the fundamental elements of analytical geometry. The study of the line is fundamental in geometry, physics and many areas of engineering. Understanding its equation and properties helps to model trajectories, analyze motions and solve optimization problems.

Algebraic representation. Equations of the line in the Cartesian plane

A line in the Cartesian plane can be described in different ways, each of which offers a different perspective depending on the information available or the purpose of the analysis. These algebraic expressions, known as line equations, allow the direction and position of a line to be accurately represented using coordinates and parameters. Among the most common forms are the vector equation, the parametric equations, the symmetric equation, the point-slope equation, the explicit equation, and the general equation. Each of these representations is equivalent to each other, but is most useful in different contexts. In the following sections we will explore each of these equations.

Vector equation of the line

The vector equation of a line is a way of representing the line in the Cartesian plane using vectors. The line is expressed as the sum of a position vector and a director vector scaled by a parameter. The vector equation of a line passing through a point P0(x₀, y₀) and having a director vector v = (a, b) is written as:

R =r₀ +tv

Where,

R = (x,y) is the position vector of any point on the line.
R0 = (x₀, y₀) is the position vector of a known point on the line.
V = (a,b) is the director vector, which indicates the direction of the line.
t is a real parameter, which allows to generate different points on the line by varying its value.

Parametric equations of the line

A line in the plane can be represented by a system of parametric equations. This form is especially useful when it comes to representing the line from an initial point and a direction. The general form of the parametric equations is:

x = x₀ + a·t
y = y₀ + b·t

Where,

(x0, y0) is a known point on the line.

a and b are the components of the directing vector of the line.

The parameter t can take any real value, and allows to generate all the points of the line. The parametric equations are obtained from the vector equation by separating its x, y components.
This equation is especially useful when working with vectors and in the description of trajectories in physics, where a parameter is needed to determine the position along the line.

Symmetric equation of the line

The symmetric equation is expressed as:

(x – x₀)/a = (y – y₀)/b

Where,

(x₀, y₀) is a known point through which the line passes.

a and b are the components of the directing vector.

The symmetric equation is easily derived from the parametric equations. This equation is useful when a point and the direction of the line are known. It is used in analytic geometry and intersection problems.

Equation point-slope of the line

The point-slope equation is expressed as:

y = y₀ + m(x−x₀)

Where,

m represents the slope

(x₀, y₀) is a known point on the line

It is obtained from the symmetric equation by mirroring the y-value. This equation is useful when a specific point on the line and its slope are known. It is used in calculations of equations of tangent lines and trajectories in physics.

Explicit equation of the line

The explicit equation of the line is expressed as:

y = mx + b

Where,

m is the slope of the line, which indicates how many units the line goes up or down for each unit it advances along the x-axis.

n is the value at which the line cuts the y-axis, that is, the point where x = 0.

This equation is a particular case of the point-slope equation in which the known point is the point where the y-axis intersects.

General equation of the line

The most general form for the equation of a line is:

Ax + By + C = 0

Where,

A, B, y C are constants.

x and y are the coordinates of any point on the line. It is obtained by rewriting the explicit or point-slope equation in a standard form.

It is obtained by rewriting the explicit or point-slope equation in a standard form. This equation is the most complete and flexible representation of a line. It allows to quickly identify if two lines are parallel or perpendicular. It is used in many algebraic calculations, such as the distance from a point to a line.