Cartesian Plane

The Cartesian Plane is a fundamental mathematical idea that allows us to visualise numerical relationships. With two intersecting number lines at a point called the origin, we can identify certain points on a flat plane using pairs of coordinates. René Descartes, a French mathematician, created this system, which bridges the gap between algebra and geometry. From simple plotting to advanced functions, the Cartesian Plane is a versatile tool that takes numbers and turns them into meaningful visual patterns.

1.0Cartesian Plane Meaning

The Cartesian Plane or Coordinate Plane is a two-dimensional grid created by the intersection of two lines perpendicular to each other: the horizontal x-axis and the vertical y-axis. The two axes intersect at the point (0, 0).

Observe that if any point on the x-axis is on the left side of the origin, the sign of the point will be negative and positive if the point is on the right side. Likewise, for the y-axis, if the point is above the origin, then it will be positive and otherwise negative.

2.0Cartesian Plane Quadrants and Coordinates

In 2D geometry, the origin or the point of intersection separates the Cartesian plane into four regions called the Cartesian plane quadrants, referred to as Quadrants I, II, III, and IV. Each quadrant varies depending on the signs of the coordinates of a point. As follows:

These quadrants begin at the top right corner of the Cartesian plane and are numbered using the counterclockwise rule. Refer to the Cartesian Plane Drawing for a better idea of these quadrants.

 Coordinates

Every point on the Cartesian Plane is represented as a coordinate pair, (x, y) for 2D and (x, y, z) for 3D, where

• x, also referred to as the abscissa, tells us how far to go left or right.
• y, also referred to most often as the Ordinate, tells us how much to move up or down.
• z is used for moving forward or backward (in or out of the screen), providing a sense of depth.

3.0Types of Cartesian Planes

The Cartesian Plane can be in various dimensions, depending on the number of axes involved. Some key types of Cartesian planes utilised and learned in mathematics are:

One-Dimensional Cartesian Plane

• Has only one axis (usually the x-axis).
• Used to graph along a straight line, like a number line.

Two-Dimensional Cartesian Plane

• The most common form used in algebra and geometry.
• Has an x and a y axis. 
• Used to graph equations, plot shapes, etc.

Three-Dimensional Cartesian Plane

• Includes a third axis: the z-axis, representing depth.
• Used in advanced math, physics, engineering, and 3D modelling.

4.0How to Use the Cartesian Plane

The Cartesian Plane helps us locate points, graph shapes, and graph equations. This is how the Cartesian Plane Drawing with coordinates is constructed:

• Every point is written as an ordered pair (x, y).
• Start at the origin (0, 0).
• Move along the x-axis (to the right if the number is positive, to the left if the number is negative).
• Then, move along the y-axis (up if the number is positive, down if the number is negative).
• Put a mark where the two paths cross.
• Plot two or more points that satisfy an equation or form a shape.
• Connect them to form lines, triangles, rectangles, or other shapes.

5.0Cartesian Plane Equation

Equations in the Cartesian Plane express the relationship between grid coordinates. Depending on whether the plane is 2D or 3D, the equations used are different.

Cartesian Plane Equation for 2D Plane

In 2 dimensions, the most familiar equations that the students will be learning are lines and curves. The equations assist one in visualising these lines by plotting them on the Cartesian plane:

• Straight Line (Linear Equation): The form of expressing a line on the Cartesian plane in 2 dimensions is:

y=m x+c

Here,  

• m is the gradient of the line and can be determined for any two given points as:

$m=\frac{y_2-y_1}{x_2-x_1}$

• c is the y-intercept.

Cartesian Plane Equation for 3D Plane

These equations, as the name indicates, describe a line or any other plane in three-dimensional space. Some of the most common 3D equations and their Cartesian plane form are given below:

• Equation of Plane: A Plane is a flat three-dimensional surface in space, which is described by the standard equation:

a x+b y+c z=d

Here, a, b, and c are components of a vector perpendicular to the plane, and d is the constant term. 

• Line (in Cartesian Form): The equation represents a line passing through the points (x₁, y₁, z₁) in 3D space: 

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$

Here, a, b, and c are the directions of the vector. 

6.0Cartesian Plane Formula

The Cartesian Plane allows us to calculate distance, midpoints, slopes, and many other things using some formulas. However, these formulas could be a bit different for 2D and 3D space. Like:

1. Distance formula: 

For two points (x₁, y₁) and (x₂, y₂) in two-dimensional cartesian plane: 

$\text{ Distance }=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

For two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in three-dimensional cartesian plane:

$\text{ Distance }=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$

2. Midpoint Formula: 

The midpoint formula for two points (x₁, y₁) and (x₂, y₂) in two-dimensional cartesian plane:

$\text{ Midpoint }=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

The midpoint formula for two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in three-dimensional cartesian plane:

$\text{ Midpoint }=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$

7.0Solved Examples on Cartesian Plane

Problem 1: Plot the point (–3, 4) on the Cartesian plane. In which quadrant does it lie?

Solution:

Based on the Cartesian plane demonstrated above and the quadrants' sign convention, (-3, 4) will be in Quadrant II. 

Problem 2: Draw the following points and mark them: A(2, 3), B(–4, 1), C(–2, –5), D(3, –3). Then, name the quadrant where each point falls.

Solution:

A(2, 3) is in Ist Quadrant

B(–4, 1) is in 2nd Quadrant

C(–2, –5) is in 3rd Quadrant

D(3, –3) lies in 4th Quadrant