Cartesian Coordinate System

The Cartesian Coordinate System is a method used to locate the exact position of a point in a plane (2D) or in space (3D).

In this system, the position of a point is represented by an ordered pair (x, y) in two dimensions or an ordered triplet (x, y, z) in three dimensions. These coordinates represent the signed distances of the point measured from the coordinate axes, meaning the values can be positive or negative depending on the direction from the origin.

Components of the Cartesian Coordinate System

The Cartesian Coordinate System is characterised by basically three components. These are Cartesian Coordinates, Coordinate Axes, and Cartesian Planes.

Cartesian Coordinates

Cartesian coordinates are numerical values used to describe the exact position of a point in a plane or in space. They tell us how far a point is located from the reference axes.

In a 2D Cartesian system, the position of a point is represented by an ordered pair (x, y).

• The x-coordinate shows the horizontal distance of the point from the y-axis and is called the abscissa.
• The y-coordinate shows the vertical distance of the point from the x-axis and is called the ordinate.
  Together, these two values are known as the coordinates of the point.

In a 3D Cartesian system, a point is represented by an ordered triplet (x, y, z).

• The z-coordinate, called the applicate, shows the distance of the point along the z-axis in three-dimensional space.

Coordinate Axes

Coordinate Axes are the reference axes that are used to measure the distance travelled by points. In 2D Coordinate System, there are two coordinate axes X and Y and in a 3D Coordinate System, there are three axes, X, Y and Z.

Cartesian Plane

The Cartesian Plane is a flat surface formed by two perpendicular lines called the x-axis and y-axis. These lines intersect at a point called the origin and divide the plane into four regions known as quadrants.

Dimension of Coordinate System

Dimension of Coordinate System basically tells about the number of points used to identify the location of a point in Coordinate Geometry. Depending on this there are three types of coordinate system, these are:

One Dimensional Coordinate System

In a one-dimensional (1D) coordinate system, the position of a point is described by a single number. The point moves along a straight line, just like on a number line.

Two Dimensional Coordinate System

In a two-dimensional Cartesian coordinate system, a point is located in a plane formed by two perpendicular axes- the x-axis and the y-axis.

The position of a point is written as an ordered pair (x, y), which represents its horizontal and vertical distances from the origin.

The two axes divide the plane into four regions called quadrants, which are numbered in an anticlockwise direction starting from the upper right quadrant, as shown in the image.

Three Dimensional Coordinate System

In a three-dimensional Cartesian coordinate system, a point is located in space using three coordinates. The position of a point is written as (x, y, z), where each value represents the distance of the point along the x, y, and z axes.

The three mutually perpendicular axes divide the space into eight regions called octants.

If a point lies on a coordinate plane, the coordinate perpendicular to that plane is zero.

For example:

• On the xy-plane, the z-coordinate is zero.
• On the yz-plane, the x-coordinate is zero.
• On the xz-plane, the y-coordinate is zero.

Mapping Ordered Pairs on the Coordinate Plane

In Cartesian System of Coordinates, to plot a point we measure the distance from the coordinate axes. For this we first need to see the coordinates. Let's say we have to plot a Point P(2, 3) then we first see that there are two coordinates hence the point need to be plotted in 2D Cartesian Plane.

To Plot the point we first observe the value of x which is 2 and travel 2 units in +x direction. Then we see that the value of y is 3, hence we will travel 3 units from your current position to the direction of +y axis. Thus this will be the require location of the point P(2, 3). Here both coordinates are positive hence the point lies in first quadrant.

In case of 3D Cartesian System of Coordinates we start with x from the origin, then move to the direction of y and then to the direction of z-axis. Thus, in case of 3D cartesian system of coordinates point is located in space if all the three coordinates are non-zero.

Formulas in Cartesian Coordinate System

This system helps in locating points and drawing graphs of algebraic functions. It also allows us to calculate the distance between points in a plane or in space.

Distance Formula

The distance between two points is given by:

• Distance Formula for Two Points in 2D: √{(x₂ - x₁)² + (y₂ - y₁)²}
• Distance Formula for Two Points in 3D: √{(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²}

Section Formula

Section formula is given to find the coordinates of a point which divides a given line in a given ratio.

Consider a line which is formed by joining two points (x₁, y₁) and (x₂, y₂) is divided by a Point P(x, y) in the ratio m:n then the coordinates will be given by

x = (mx₂ + nx₁)/(m + n) and y = (my₂ + ny₁)/(m + n)

Mid-Point Formula

In case of section formula if the ratio becomes equal i.e. 1:1 then it is called Midpoint Formula. Hence, if a Point is mid-point of a line then its coordinates are given as

x = (x₁ + x₂)/2 and y = (y₁ + y₂)/2

Slope of a Line

Slope of a line is the inclination of line with respect to the coordinate axes. The slope of a line is calculated as m = Tan θ where θ is the angle between line and the coordinate axis.

The formula for slope of line in cartesian form is given as

m = (y₂ - y₁)/(x₂ - x₁)

We know that Cartesian Coordinate System can also be used to draw graph for various algebraic expressions. In this article we will learn Cartesian Coordinate Equation of line and plane.

Equation of Line in Cartesian Form

The standard equation of a line is given by a linear equation expressed as ax + by + c = 0. However there are other forms also in which the equation of a line can be given. These equation are mentioned below:

• Slope-Intercept Form of Line: y = mx + C where m is the slope and C is the intercept.
• Intercept Form of Line: x/a + y/b = 1
• Point-Slope Form of Line: (y - y₁) = m(x - x₁)
• Two Point Form of Line: (y - y₁) = {(y₂ - y₁)/(x₂ - x₁)}(x - x₁)
• Normal Form of Line: L = xcos θ + ysin θ = P

Equation of Plane in Cartesian Form

A plane is a two dimensional flat region bounded by two coordinate axes. The different equations of Plane in cartesian form is given as follows:

Equation of Plane in Normal Form: \vec r . \hat n = d where d is the perpendicular distance from the origin and n is the unit vector on the plane.

Equation of Plane Passing through three Non Collinear Points: (\vec r - a)[(\vec b - \vec a)\times(\vec c - \vec a)] = 0 where a, b and c are non-collinear points.

Equation of Plane passing through intersection of Two Planes: If a plane pass through through intersection of two planes whose equation is given as \vec r . \hat n_1 = d_1 and \vec r . \hat n_2 = d_2 then its equation is given as \vec r ( \hat n_1 + \lambda \hat n_2) = d_1 + \lambda d_2

Cartesian Representation of Complex Numbers

We know that a complex number is given as Z = a + ib where a is the real part and ib is the imaginary part. The letter 'i' stand for iota whose value is equal to √-1 which is an imaginary number as there exists no number whose square is a negative number. This is from the law of exponents that any number raised to even power results in positive number always.

In general to represent a number in Cartesian System of Coordinates we take the both axis to be real i.e. numbers on the coordinate axes are real numbers. However to plot a complex number one of the axis usually x-axis is the real axis and the other axis i.e. the y-axis is the imaginary axis. A pictorial representation of plotting of Complex Number is given below:

Application of Cartesian Coordinate System

The Cartesian coordinate system is used in many areas.

• It helps in finding locations on Earth using latitude and longitude. The same idea is used in maps and navigation systems.
• In engineering and construction, it is used to measure distances and draw lines or curves.
• It is also used to draw graphs of algebraic expressions and to understand the relationship between different variables.

People Also Read:

• Coordinate Geometry
• Linear Equation in One Variable
• Polynomial
• Geometry

Cartesian Coordinates System Examples

Example 1: Find the coordinate of point O(x, y) which divides the line joining the points P(3, 4) and Q(1, 2) in the equal ratio.

It is given that O divides PQ in equal ratio. Hence, O is the midpoint of PQ. Therefore by using midpoint formula we have

x = (3 + 1)/2 and y = (4 + 2)/2

⇒ x = 4/2 = 2 and y = 6/2 = 3

Hence the coordinates of the point is O(2, 3)

Example 2: Find the slope of the line formed by joining the points (3, 2) and (-3, -2).

The slope of a line is given by the formula

m = (y₂ - y₁)/(x₂ - x₁)

⇒ m = (-2 - 2)/(-3 - 3)= -4/-6 = 2/3

Example 3: Find the distance between the two points A(-2, 3) and B(3, 1).

Here we see that each point is indicated by two numbers. Hence this is the case of two dimensional coordinate system.

Distance between two points in is given as √{(x₂ - x₁)² + (y₂ - y₁)²}

⇒ AB = √{(3 - (-2))² + (1 - 3)²} = √{(5)² + (-2)²} = √29 units

Example 4: Find the distance of the points A(2, -1, 4) from the origin.

Here the point is indicated by three values hence this is a case of 3D Cartesian Coordinate System. In 3D Cartesian Coordinate System, the distance of the point from the origin is given as √(x² + y² + z²)

Hence OA = √{(2)² + (-1)² + (4)²} = √21 units

Cartesian Coordinate System Questions

Q1: Find the distance between Origin and Point P(-3, -2)

Q2: Find the slope of the line joining the points (-1, 4) and (2, -3)

Q3: Find the equation of a line using slope form of a line which passes through point (3, 4) and slope is 2/3.

Q4: Find the coordinates of a point which is the midpoint of a line joining the points (1, 3) and (-3, 4).

Q5: Locate Points (-5, 6), (2, -3), (1, 2) and (-1, 0) in Cartesian System.