Cartesian Plane

Cartesian plane is defined as the two-dimensional plane used in the Cartesian coordinate system. This plane is formed by intersecting two perpendicular lines called the x-axis and the y-axis and their intersection is called the origin. This method of distributing the 2-Dimensional space into four areas was first introduced by Rene Descartes in the early 17th century. We can specify any point in this cartesian system by an ordered pair. This order pair is sufficient for telling the position of any point with respect to the origin of the Cartesian system.

In this article, we will learn about, the cartesian plane, parts of the cartesian plane, locating any point in the cartesian, and others.

Cartesian Plane Definition

A two-dimensional coordinate plane system that is formed by intersecting two perpendicular lines is called the cartesian plane. The two intersecting lines are called the x-axis and y-axis respectively. Generally, we define the Horizontal line as the x-axis and the vertical line as the y-axis. The point of intersection of these two lines is called the Origin of the Cartesian system, and it represents the starting point of the Cartesian plane and all the points in the plane are measured with respect to this point.

A point in the cartesian plane is an ordered pair in (x, y) which represent the position of the point in the cartesian system with respect to the origin. The complete 2-D plane is divided into four equal areas in this coordinate system and these areas are called the quadrants of the cartesian plane.

We mark the x-axis and the y-axis in the cartesian plane using positive and negative numbers. We use proper sign convention to easily mark the cartesian plane,

• On moving the right-hand side on the horizontal (x) axis from the origin we mark it as the positive direction.
• On moving the left-hand side on the horizontal (x) axis from the origin we mark it as the negative direction.
• On moving up on the vertical (y) axis from the origin we mark it as the positive direction.
• On moving down on the vertical (y) axis from the origin we mark it as the negative direction.

Cartesian Plane Example

The image below shows Cartesian Plane:

Parts of a Cartesian Plane

Various parts of the cartesian plane are,

• Axes
• Origin
• Quadrants
• Coordinate of a Point

Let's understand these parts of the Cartesian Plane in detail.

Axes

Cartesian plane is made by intersecting two mutually perpendicular lines, one horizontal line, and one vertical line these are called the axes. We have two axes in the cartesian plane that are,

• x-axis
• y-axis

The x-axis is the horizontal axis whereas the y-axis s the vertical axis.

Origin

The intersection point of the x-axis and the y-axis is called the origin of the cartesian plane. It is denoted by an ordered pair (0, 0)

Cartesian Plane Quadrants

The two intersecting axes X-axis and Y-axis divide the Cartesian plane into four equal parts and these four parts are called the quadrants of the Cartesian Plane. We denote these quadrants as,

• First (I) Quadrant
• Second (II) Quadrant
• Third (III) Quadrant
• Fourth (IV) Quadrant

The image added below shows the axes, origin, and quadrants of the Cartesian Plane.

Coordinate of a Point

Coordinate of a point is the address of the point in any space (one, two, or three-dimensional), which helps to locate a point in space. For one, dimensional space, coordinates are represented by any real number which gives the distance of the point from the origin i.e., 0.

For two-dimensional space, the coordinates are represented by the ordered pair (x, y) where x and y are referred to as the abscissa and ordinate.

• Abscissa: The distance of a point from the x-axis is called abscissa.

• Ordinate: The distance of a point from the y-axis is called ordinate.

For example, a point with coordinate (1, 3) represents a point that is at 3 unit distance from the y-axis and 1 unit distance from the x-axis.

Plotting Points on Cartesian Plane

We can easily plot any point on the Cartesian plane. The ordered pair (x, y) is represented in the cartesian plane. For example, we can represent points with coordinates (2,3) using the following steps in the Cartesian Plane. 

Step 1: Take the ordered pair (2, 3)

Step 2: Take point 2 on the right-hand side of the x-axis and draw a line passing through this point., i.e. passing through x = 2 which is parallel to the y-axis.

Step 3: Take point 3 on the upward of the y-axis and draw a line passing through this point., i.e. passing through y = 3 which is parallel to the x-axis.

Step 4: The intersection of the line passing through x = 2 and y = 3 is the required point (2, 3)

A point (x, 0) lines on the x-axis, and the point (0, y) lie on the y-axis.

Cartesian Plane Graph

In the cartesian plane we have two variables the x and the y and the relation between these variables is called the equations. We can easily plot the graph of these equations, the graph of these equations is all the points (x, y) that satisfies the given equation.

We can easily plot the graph of any equation using the steps discussed below,

Step 1: Take some random numerical value for the x.

Step 2: Put this value in the given equation and then find the respective value of y. Generating the ordered pair (x, y)

Step 3: Repeat steps 1 and 2 several times to get different ordered pairs of (x, y)

Step 4: Plot all the points obtained in the Cartesian plane.

Step 5: Draw a curve joining all the points. 

Thus, we can easily plot the graph of some well-known curves. Some of the common curves are,

• x + y = a (this represents a straight line)
• x² + y² = a (this represents a circle)
• x² + y = a (this represents a parabola)

We can draw the graph of the curves given above using this method.

One Dimensional Plane(Line)

In one dimension we draw a line and choose the origin of that line accordingly and now the position on any point in one dimension is measured with respect to the origin in that line.

We denote the right side of the origin in the line by the positive numbers and the left side of the origin in the line by the negative numbers. The line is called the number line any number can be plotted on this number line.

Three Dimensional Plane

In 3-Dimensional space, we require 3 mutually perpendicular axes that divide the 3-D space into 8 equal spaces. The three axes are the x-axis, the y-axis, and the z-axis. We live in 3-D so we can imagine that the space around us is 3-D space, and the position of any point in this 3-D space is given by the ordered pair (x, y, z)

Cartesian Representation of Complex Numbers

Complex numbers represent all numbers including real numbers and imaginary numbers. They are represented using the symbol Z. The general form of the complex number is,

Z = a + ib

Where i is the iota.

They can be easily represented on the Cartesian plane. If we take one axis of the cartesian plane as the axis of the real number and the other axis of the cartesian plane as the axis of imaginary numbers then their position in this plane represents the complex number. This plane is called the complex plane.

For example, the complex number 3 - 2i is represented in the image below,

How to Plot Complex Numbers in Cartesian Plane?

The complex number in the Cartesian plane can be easily plotted. If the given complex number is a+ib then we take the x-coordinate as a and the y-coordinate as b and thus the ordered formed is (a, b). 

We can easily plot the complex number  using the steps discussed below,

Step 1: Mark the given complex number in the standard form a+ib, i.e. if the complex number is given as a-ib write it as, a+i(-b)

Step 2: Mark the ordered pair (a, b) according to the given complex number.

Step 3: Plot the ordered pair (a, b) using the step discussed above. This gives the position of the required complex number a+ib in the complex plane.

Read, More

• Parallel Lines
• Coordinate Geometry
• Distance Formula

Solved Examples on Cartesian Plane

Example 1: Plot the point (2, 3) on the cartesian plane.

Solution:

Given points (2, 3)

Here,

• x-coordinate = 2
• y-coordinate = 3

It can be plotted on the cartesian plane as,

Example 2: Plot the point (-3, 4) on the cartesian plane.

Solution:

Given points (-3, 4)

Here,

• x-coordinate = -3
• y-coordinate = 4

It can be plotted on the cartesian plane as,

Example 3: Plot the point (-1, -2) on the cartesian plane.

Solution:

Given points (-1, -2)

Here,

• x-coordinate = -1
• y-coordinate = -2

It can be plotted on the cartesian plane as,