Cartesian Coordinate System

The cartesian coordinate system is a branch of mathematics that tells about how to represent a point uniquely in the n-dimensional coordinate plane. The theory of the cartesian system was proposed by a French philosopher and mathematician called Rene Descartes in the 17th century. This cartesian coordinate system provided the relationship between Euclidean geometry and algebra, which has revolutionized the study of mathematics. The cartesian coordinate system is the foundation of analytical geometry and helps in the representation of lines, curves, geometric figures in the n-dimensional plane. Let us learn more about the cartesian system and the terms associated with it.

The system which we use to label points in a plane is known as the Cartesian System. The cartesian form is derived from the number line. To understand the cartesian coordinate system we must know about the number line thoroughly. In this system, we have the following defined parameters such as:

• Two perpendicular lines are named as X-axis and Y-axis.
• The plane is called the Cartesian, or coordinate plane and the two lines X and Y when put together are called the coordinate axes of the system.
• The two coordinate axes divide the plane into four parts called quadrants.
• The intersection point of the axes is the zero of the Cartesian System. This point will generally be denoted by O. The coordinates of the origin are denoted as (0, 0).
• To specify the position of any point P in the plane, we measure the distance x we have to move along X, and then the distance y we have to move parallel to Y, to reach from O to P. Distances can be negative.
• For example, if you have to move right, then x will be positive. Similarly, if you have to move down on Y, then y will be negative.
• The two real numbers x and y plotted together will describe the position of P uniquely. We can write this as follows: P = (5, 6) [from the below figure]. Thus, the location of P can be labeled uniquely by two real numbers. For different positions of P, this pair of real numbers will be different.

Now observe the following graphical representation of cartesian coordinates and read the above description again.

Let us discuss a few parameters associated with the cartesian system to have a basic understanding of how well we can read the coordinates.

Cartesian Coordinates

In the cartesian system, the x-coordinate of a point is its perpendicular distance from the y-axis. It is measured along the x-axis which is positive along the positive direction and negative along the negative direction. For point P, it is +5 on the positive x-axis. This x-coordinate is called the abscissa.

In the cartesian system, the y-coordinate of a point is its perpendicular distance from the x-axis. It is measured along the y-axis. For point P, it is +6 on the positive y-axis. This y-coordinate is called the ordinate.

In the cartesian coordinate system we generally start the bifurcation with one dimension, then two-dimension, and then a three-dimensional system. Let us discuss these three cartesian systems dimensions in detail.

One Dimensional Cartesian Coordinate System

The cartesian coordinate system for a one-dimensional space is a straight line having the origin O and a positive side and a negative side of the line. One-dimensional means either the plane have a horizontal line or a vertical line. If the line is horizontally plotted then the right side is taken as positive and the left side is taken as negative. Whereas, if the line is oriented vertically then the upper part of the line is taken as positive and the lower part of the line is taken as negative.

Each point on the line is specified with reference to the origin, and with a defined scale. The coordinate of the point is prefixed with a + or - sign and the numeric value to represent its distance from the origin O. Generally the one-dimensional line is referred to as the number line and any of the real numbers can be conveniently represented on this number line.

Two Dimensional Cartesian Coordinate System

A cartesian plane divides the plane space into two dimensions and is useful to easily locate the points. It is also referred to as the coordinate plane. The two axes of the coordinate plane are the horizontal x-axis and the vertical y-axis. These coordinate axes divide the plane into four quadrants, and the point of intersection of these axes is the origin (0, 0). Further, any point in the coordinate plane is referred to by a point (x, y), where the x value is the position of the point with reference to the x-axis, and the y value is the position of the point with reference to the y-axis. The coordinates of the point in the first quadrant are (+x, +y), the second quadrant is (-x, +y), the third quadrant is (-x, -y), and the fourth quadrant is (+x, -y).

Three Dimensional Cartesian Coordinate System

The three-dimensional cartesian coordinate system consists of three axes, the x-axis, the y-axis, and the z-axis, which are mutually perpendicular to each other and have the same units of length across all three axes. Similar to the two-dimensional coordinate system, where the point of intersection of these three axes is the origin O, and these axes divide the space into eight octants. Any point in space is represented with the coordinates (x, y, z). The x value of the point (x, y, z) is referred to as the abscissa, the y value of the point is referred to as the ordinate and the z value is referred to as applicate.

Further the coordinates of a points in the eight octants are represented as (+x,+y,+z), (-x,+y,+z), (+x,+y,-z), (-x,+y,-z), (+x,-y,+z), (-x,-y,+z), (+x,-y,-z), (-x,-y,-z).

Please note, the cartesian coordinate system can have an n-dimensional system to represent numerous quantities at once. But the higher dimensions cannot be presented geometrically and assumed in theory only. The higher dimensional systems have major applications in computer programming and artificial intelligence.

The formulas of the cartesian coordinate system help in conveniently proving the various properties of lines, curves, planes in a two-dimensional and three-dimensional system. The formulas of the cartesian coordinate system include the distance formula, slope formula, midpoint formula, section formula, equations of a line in two and three dimensions, equations of curves, and equations of a plane. Let us know more about each of the formulas in the below paragraphs.

Cartesian Coordinates Distance Formula

The distance between two points \((x_1, y_1)\) and \(x_2, y_2) \) is equal to the square root of the sum of the squares of the difference of the x coordinates and the y-coordinates of the two given points. The formula for the distance between two points is as follows.

D = \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Slope Formula

The slope of a line is the inclination of the line. The slope can be calculated from the angle made by the line with the positive x-axis, or by taking any two points on the line. The slope of a line inclined at an angle θ with the positive x-axis is m = Tanθ. The slope of a line joining the two points \((x_1, y_1)\) and \(x_2, y_2) \) is equal to m = \( \frac {(y_2 - y_1)}{(x_2 - x_1)} \).

Mid-Point Formula

The formula to find the midpoint of the line joining the points \((x_1, y_1)\) and \(x_2, y_2) \) is a new point, whose abscissa is the average of the x values of the two given points, and the ordinate is the average of the y values of the two given points. The midpoint lies on the line joining the two points and is located exactly between the two points.

\((x, y) =\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\)

Section Formula

The section formula is useful to find the coordinates of a point that divides the line segment joining the points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m : n\). The point dividing the given two points lies on the line joining the two points and is available either between the two points or on the line, beyond the two points.

\((x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) \)

Cartesian Equation of a Line

This equation of a line represents all the points on the line, with the help of a simple linear equation. The standard form of the equation of a line is ax + by + c= 0. There are different methods to find the equation of a line. Another important form of the equation of a line is the slope-intercept form of the equation of a line (y = mx + c). Here m is the slope of the line and c is the y-intercept of the line. Further, the other forms of the equation of a line are point-slope form, two-point form, intercept form, and the normal form. The differential equations of a line are as follows.

• Point Slope Form: (y - y\(_1\)) = m(x - x\(_1\))
• Two Point Form: \((y -y_1) = \frac{(y_2 - y_1)}{(x_2 - x_1)}(x - x_1) \)
• Slope Intercept Form: y = mx + c
• Intercept Form: \(\frac{x}{a} + \frac{y}{b} = 1 \)
• Normal Form: xcosθ + ysinθ = P

Cartesian Equation of a Plane

The equation of a plane in a cartesian coordinate system can be computed through different methods based on the available inputs values about the plane. The following are the four different expressions for the equation of plane.

• Normal Form: Equation of a plane at a perpendicular distance d from the origin and having a unit normal vector \(\hat n \) is \(\overrightarrow r. \hat n\) = d.
• Through three Non Collinear Lines: The equation of a plane passing through three non collinear points \(\overrightarrow a\), \(\overrightarrow b\), and \(\overrightarrow c\), is \((\overrightarrow r - \overrightarrow a)[(\overrightarrow b - \overrightarrow a) × (\overrightarrow c - \overrightarrow a)] = 0\).
• Intersection of Two Planes: The equation of a plane passing through the intersection of two planes \(\overrightarrow r .\hat n_1 = d_1\), and \(\overrightarrow r.\hat n_2 = d_2 \), is \(\overrightarrow r(\overrightarrow n_1 + λ \overrightarrow n_2) = d_1 + λd_2\).

☛Important Notes on Cartesian Coordinate System

• The point of intersection of both the axes is known as the origin and its coordinates are (0, 0).
• There can be an infinite number of points on a cartesian coordinate plane.
• Points that lie on any of the number lines do not belong to any quadrant.
• A point that is above the x-axis has its y-coordinate positive and if the point lies below the x-axis, then its y-coordinate is negative.
• A point that lies to the right of the y-axis has its x-coordinate positive and if the point lies to the left of the y-axis, then the x-coordinate in negative.

☛Related Topics

The following topic would be helpful in better understanding of the cartesian coordinate system.

• Coordinate Plane
• Plane Definition
• Polar Coordinates
• Cartesian Coordinates
• Coordinate Geometry
• Equation of a Plane

What Is Meant By Cartesian Coordinate System?

The cartesian coordinate system is a system with gives reference axes to represent points, lines, curves, planes. The algebraic equations can be represented geometrically using the cartesian coordinate system. The cartesian coordinate systems is of one dimension, two dimensions, three-dimension, and n dimension. The points in a cartesian coordinate system are expressed as (x, y), or (x, y, z).

What Is the Cartesian Coordinate System Used For?

The cartesian coordinate system can be used to represent points, lines, curves, planes. The vectors can be represented in a three-dimensional cartesian coordinate system. Calculus, trigonometry, algebra, probability uses the cartesian coordinate system to geometrically represent the mathematical expressions.

How Do We Represent a Point in a Cartesian Coordinate System?

The point in a cartesian coordinate system is expressed as (x, y), (x, y, z), \((x_1, x_2, x_3, ...x_n)\). The x coordinate is the abscissa, the y coordinate of the point is point is called the ordinate, and the z coordinate of the point is called the aplicate.

How Do We Represent a Line in a Cartesian Coordinate System?

This equation of a line in a cartesian coordinate system is represented by a simple linear equation. The standard form of the equation of a line is ax + by + c= 0. There are different methods to find the equation of a line. Another important form of the equation of a line is the slope-intercept form of the equation of a line (y = mx + c). Here m is the slope of the line and c is the y-intercept of the line. Further, the other forms of the equation of a line are point-slope form, two-point form, intercept form, and the normal form. The different forms of equations of a line in a cartesian coordinate system are as follows.

• Point Slope Form: (y - y\(_1\)) = m(x - x\(_1\))
• Two Point Form: \((y -y_1) = \frac{(y_2 - y_1)}{(x_2 - x_1)}(x - x_1) \)
• Slope Intercept Form: y = mx + c
• Intercept Form: \(\frac{x}{a} + \frac{y}{b} = 1 \)
• Normal Form: xcosθ + ysinθ = P

How Do You Represent A Conic In A Cartesian Coordinate System?

The important conics in the cartesian coordinate system are the circle, parabola, ellipse, and hyperbola. The following are the equations of each of these comics in a cartesian coordinate system.

• Equation of a Circle: (x - a)² + (y - b)² = r²
• Equation of a Parabola: y² = 4ax
• Equation of an Ellipse: x²/a² + y²/b² = 1
• Equation of a Hyperbola: x²/a² - y²/b² = 1

How Do You Represent A Line in A Three Dimensional Cartesian Coordinate System?

The equation of a line in a three-dimensional cartesian coordinate system can be computed from the following two methods. The two methods of finding the equation of a line is as follows.

• The equation of a line passing through a point and parallel to a given vector: r = a + λb
• Equation of a line passing through two given points: r - a = λ)b - a)

How Do We Represent a Plane in a Cartesian Coordinate System?

The equation of a plane in a cartesian coordinate system can be computed through different methods based on the available inputs values about the plane. The following are the four different expressions for the equation of plane.

• Normal Form: Equation of a plane at a perpendicular distance d from the origin and having a unit normal vector \(\hat n \) is \(\overrightarrow r. \hat n\) = d.
• Perpendicular to a given Line and through a Point: The equation of a plane perpendicular to a given vector \(\overrightarrow N \), and passing through a point \(\overrightarrow a\) is \((\overrightarrow r - \overrightarrow a). \overrightarrow N = 0\)
• Through three Non Collinear Lines: The equation of a plane passing through three non collinear points \(\overrightarrow a\), \(\overrightarrow b\), and \(\overrightarrow c\), is \((\overrightarrow r - \overrightarrow a)[(\overrightarrow b - \overrightarrow a) × (\overrightarrow c - \overrightarrow a)] = 0\).
• Intersection of Two Planes: The equation of a plane passing through the intersection of two planes \(\overrightarrow r .\hat n_1 = d_1\), and \(\overrightarrow r.\hat n_2 = d_2 \), is \(\overrightarrow r(\overrightarrow n_1 + λ \overrightarrow n_2) = d_1 + λd_2\).