Cartesian Coordinate System
Last Updated :
10 Jun, 2024
Cartesian Coordinate System in Maths is a division of coordinate geometry where the location of a point in a plane or space is marked by a pair of numbers or three numbers. The branch of geometry that deals with the Cartesian Coordinate System is called Coordinate Geometry. The numbers which are used to identify the location of a point in a plane or space are called coordinates.
The concept of the Cartesian Coordinate System is fundamental for class 9 and class 10 students which will later help them to understand various graphs and solve problems in physics.
In this article, we will study about the Cartesian Coordinate System, cartesian coordinates, coordinate axes, two dimensional as well as three-dimensional coordinate systems in detail.
What is Cartesian Coordinate System in Maths?
The branch of Geometry that deals with the labeling of points in space or plane and marking their location with numbers is called the Cartesian Coordinate System. The geometry dealing with the Cartesian Coordinate System is also called Coordinate Geometry. The Cartesian Coordinate System helps draw graphs of various functions and hence is the fundamental concept for the interpretation of analytical geometry and various other branches of mathematics such as linear algebra, calculus, complex analysis, etc.
History of Cartesian Coordinate System
The credit for the development of the Cartesian Coordinate System goes to the French Mathematician Rene Descartes. He published the idea of the Cartesian Coordinate System in 1637. Although several others also worked on this such as Pierre De Fermat who also worked independently on the Cartesian Coordinate System but did not publish his results and findings. Another person, Nicole Orseme who was a French cleric also worked on this before Fermat and Descartes.
The thing that was common to both Descartes and Fermat was that they both used a single axis to represent the variable length. The concept of a pair of coordinates was introduced later. After the discovery of the third dimension, several new coordinate systems such as the Spherical Coordinate System, Polar Coordinate System and Cylindrical Coordinate System also developed.
Components of the Cartesian Coordinate System
The Cartesian Coordinate System is characterized by basically three components. These are Cartesian Coordinates, Coordinate Axes, and Cartesian Planes. Let’s learn them in detail.
Cartesian Coordinates
Cartesian Coordinates are the set of points that are used to identify the location of a point in cartesian space. In a 2D cartesian system, the coordinate of a point is given by an ordered pair (x, y) where x and y indicate the distance traveled in the direction of the respective axis. The x-coordinate is called abscissa and the y-coordinate is called ordinate and combinedly they are called coordinates. In 3D Coordinate Geometry, a point is identified by three points (x, y, z) where x, y, and z are the distance traveled by points in the direction of the x, y, and z axes respectively.
- Abcissa: The value of the x-coordinate is called abcissa. It tells the distance traveled by a point away from the y-axis in the direction of the x-axis in a 2D Cartesian Plane.
- Ordinate: The value of y-coordinate is called ordinate. It tells the distance traveled by a point away from the x-axis in the direction of the y-axis in a 2D Cartesian Plane.
- Applicate: The value of z-coordinate in a 3D Cartesian System of Coordinates is called ordinate. It tells the distance traveled by a point away from the x-axis and y-axis in the direction of z-axis in 3D 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 3D Coordinate System there are three axes X, Y and Z.
Note: All the axes intersect each other a point called origin whose coordinate is given as (0, 0) in 2D coordinate and (0, 0, 0) in 3D cartesian coordinate system.
Cartesian Plane
Cartesian Plane is a 2D space or region bounded a pair of coordinate axes in which a point moves. In 2D system there is only XY plane which is divided into quadrants by the X and the Y-axis. In 3D system there are three Cartesian Planes namely XY Plane, YZ plane and XZ plane bounded by the axis that are present in their names.
Dimension of Coordinate System
Dimesion 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
- Two Dimensional Coordinate System
- Three Dimensional Coordinate System
One Dimensional Coordinate System
In 1D Coordinate System the location of a point is given using one point. In this case the point travels along a line instead of space or a plane.
It is similar to representing a number on the number line. This system was used by Descartes and Fermat to give the concept of Cartesian Coordinate System.
Two Dimensional Coordinate System
In two dimensional Cartesian Coordinate system, the point travels either in plane bounded by two cartesian coordinate axes. In 2D Cartesian Coordinate System the location of a point is given using a pair of numbers called coordinates expressed as (x, y) where x is called abcissa.
It indicates the distance traveled in the direction of x-axis traveled away from y axis. The value of y is called ordinate and it indicates the distance traveled in y-direction away from the x-axis.
In 2D Coordinate system, the coordinate axes divides the cartesian plane in four parts where each part is called quadrant. The quadrants are marked in anticlockwise. The coordinates are in the form of (x, y) for first, (-x, y) for second, (-x, -y) for third and (x, -y) for the fourth quadrant.
The image attached below describes the quadrants and the pattern of coordinates of the points located in them.
Three Dimensional Coordinate System
In 3D Cartesian Coordinate system a point is located in space and its location is given using three points called Coordinates. In three dimensional coordinate system, a point is represented as (x, y, z) where x, y and z represents the distance traveled by points in the direction of respective axis.
The value of x is called abcissa, value of y is called ordinate and the value of z is called applicate. In 3D coordinate system the space is divided into 8 parts each called octants. The coordinates of the points located in these octants are given as (x, y, z), (-x, y, z), (x, -y, z), (x, y, -z), (-x, -y, z), (x, -y, -z), (-x, y, -z) and (-x, -y, -z).
In 3D coordinate system, if a point is on a coordinate axis then the value of that coordinate is non zero and rest two are zero. For Example, in (3, 0, 0), the point is located on x-axis at 3 units from the origin. If the point is located in a plane then the value of those coordinates are non-zero and the third coordinate is zero.
The image showing a point in 3D Cartesian System is attached below:
How to Plot Points in Cartesian System of Coordinates?
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.
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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.
We know that Cartesian Coordinate System is used to locate points and draw graphs for various algebraic function. Hence, the distance between the points and the equations for the graphs can be written using Cartesian System.
Distance Formula is used to calculate distance between two points, two lines, between a point and a line and many more. The most commonly is used to calculate distance between two points in 2D and as well as three 3D. These formulas are mentioned below:
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)
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.
The standard equation of a line is given by a linear equation epressed 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:
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: [Tex]\vec r . \hat n = d
[/Tex] 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: [Tex](\vec r – a)[(\vec b – \vec a)\times(\vec c – \vec a)] = 0
[/Tex] where a, b and c are non-collinear points.
Eqution of Plane passing through intersection of Two Planes: If a plane pass through through intersection of two planes whose equation is given as [Tex]\vec r . \hat n_1 = d_1
[/Tex] and [Tex]\vec r . \hat n_2 = d_2
[/Tex] then its equation is given as [Tex]\vec r ( \hat n_1 + \lambda \hat n_2) = d_1 + \lambda d_2
[/Tex]
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
Cartesian Coordinate System is used in variety of applications. These are mentioned below:
- Cartesian Coordinate System is used to identify location on Earth with the help of latitudes and longitudes. The same concept is used to locate any place on google maps.
- In Engineering especially architecture and civil engineering, Cartesian Coordinate System is used to calculate the distance between places, the line joining the points, the points on the curves and many more.
- Cartesian Coordinate System is used to visualize the algebraic expression by drawing the graph and establish relationship between various variables.
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Cartesian Coordinates System Examples
Example 1: Find the distance between the two points A(-2, 3) and B(3, 1)
Solution:
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 2: Find the distance of the points A(2, -1, 4) from the origin
Solution:
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
Example 3: 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.
Solution:
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 4: Find the slope of the line formed by joining the points (3, 2) and (-3, -2)
Solution:
The slope of a line is given by the formula
m = (y2 – y1)/(x2 – x1)
⇒ m = (-3 – 3)/(-2 – 2) = -6/-4 = 2/3
Cartesian Coordinate System Class 9
Cartesian Coordinate System is a very important chapter for class 9. Class 9 marks the introduction to graphing linear equations and functions. The Cartesian Coordinate System provides students with the fundamental framework for graphing these equations and understanding the relationship between variables visually.
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.
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