Cartesian Coordinates

John wants to meet his friend, Chris, who lives in another town. John starts from his house but does not know the address. Do you think it will be easy for John to reach his friend's house? Definitely, no! If John is given the right address, he can reach there easily. The same applies to a cartesian coordinate plane. If we want to locate a point on a cartesian coordinate plane, we must know the exact location of the point. The location of the point is known as its cartesian coordinates or coordinates of the point.

google maps using cartesian coordinates

Have you ever wondered about the uses of cartesian coordinates or the cartesian plane? They have several uses; for example, Google maps use the cartesian coordinate plane to show the locations.

In this short lesson, you will learn all about the cartesian coordinate plane. You will learn how to locate a point and plot a point in the different quadrants of a cartesian coordinate plane, and solve problems based on the cartesian coordinate system.

Let's begin!

Lesson Plan

What Is a Cartesian Coordinate Plane?

A cartesian coordinate plane is a system of two perpendicular number lines on which we locate points.

Cartesian Plane

The cartesian coordinate plane consists of two perpendicular lines, x-axis and y-axis.

The horizontal line is called the x-axis.

The vertical line is called the y-axis.

The intersection point of these two axes is called origin and is represented by \(O\).

The axes split through the origin, which is marked as 0 on the graph.

The side on the right of 0 is positive, while everything on the left is negative.

Similarly, as you go higher, the numbers are positive, while the negative numbers are written below 0

The positive x-axis has positive numbers such as 1, 2, 3, 4,…..

The negative x-axis has negative numbers such as -1, -2, -3, -4,…..

The positive y-axis has positive numbers such as 1, 2, 3, 4,…..

The negative y-axis has positive numbers such as -1, -2, -3, -4,…..

A cartesian coordinate plane can simply be called a coordinate plane or a plane.

It represents a 2-dimensional (2D) plane.

The cartesian coordinate system is sometimes also referred as the cartesian coordinate graph.


Cartesian Coordinates

A cartesian coordinate plane can have infinite points on it, but each point has its own unique coordinates.

Points on a cartesian coordinate plane

The red dots in the cartesian plane above are the cartesian points, and each point has unique coordinates known as cartesian coordinates.


Cartesian Coordinate on a Plane

How will you locate the coordinates of a point on a plane?

How will you find the coordinates by looking at a point?

We will find answers to these two questions in this section.

Consider a point \(P\) on the coordinate plane.

Point on a cartesian plane

There are two numbers written within the brackets.

This pair of numbers (3, 2) are known as the coordinates of the point \(P\) or we can say that this is the location of the point \(P\) on the cartesian plane.

The coordinates of point \(P\) are:

Representation of a point on a Cartesian plane

This is how we write the coordinates of a point \((x, y)\) i.e. (value on the x-axis, the value on the y-axis).

Use the simulation given below to observe how the coordinates change when the point is moved across the graph.

 
tips and tricks
Tips and Tricks
  1. Any point that lies on the x-axis has its y-coordinate zero.
  2. Any point that lies on the y-axis has its x-coordinate zero.
  3. For any line which is parallel to the y-axis, the x-coordinate of any point on the line always remains the same.
  4. For any line which is parallel to the x-axis, the y-coordinate of any point on the line always remains the same.

Coordinate Plane Quadrants on a 2-D Plane

The two axes x-axis and y-axis divide the graph into four parts.

These four parts are known as coordinate plane quadrants or we can simply call them quadrants.

Cartesian plane graph


Coordinates of Points on Coordinate Plane Quadrants

If we divide the points into the four quadrants, then the sign of the points will vary according to the quadrant in which it lies.

For example, point (3, -2) lies in the 4th quadrant.

Let’s see how to identify this.

Sign of the Points in the four quadrants

important notes to remember
Important Notes
  1. The point of intersection of both the axes is known as origin and its coordinates are (0, 0).
  2. There can be infinite number of points on a cartesian coordinate plane.
  3. Points that lie on any of the number lines do not belong to any quadrant.
  4. A point which 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.
  5. A point which 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.

Cartesian Coordinate Plane of a Three-Dimensional Space

Let's recall the scenario we introduced in the beginning about John going to visit his friend, Chris.

Let's say Chris lives in an apartment.

John arrives at the building by following the location coordinates.

But he is not sure about the floor on which Chris's apartment is.

Chris informs John his floor number, so John is finally able to reach his apartment.

This scenario is similar to how we deal with coordinate geometry in real life which is a three-dimensional world.

We need another axis.

The third axis is known as z-axis.

3-D geometry, Z axis

All three axes are perpendicular to each other and the point of intersection of all the three axes is called the origin.

These axes are nothing but number lines that expand in both directions up to infinity.


Coordinates of a Point on a Three-Dimensional Plane

Consider a point \(Q\) on a 3-D plane.

The coordinates of the points are (4, 2, 3).

This indicates the values of the point on (x-axis, y-axis, z-axis).

point on a 3d geometry axis

Representation of a point on a 3d plane


Solved Examples on Cartesian Coordinates

Example 1

 

 

William wants to throw a ball into a bucket which is placed on the coordinates (3, 10).

If William is standing on the coordinates (3, 5), how far will he have to throw the ball?

Solution

To calculate how far William will have to throw the ball, we need to calculate the distance between William and the bucket.

The x-coordinate of William's location and the bucket is 3

Thus, they are on a line parallel to the y-axis.

Hence, the distance between William and the bucket can be calculated as the difference of the y-coordinates of both the points.

\[10 - 5 = 5 units\]

Solution of cartesian plane, coordinate geometry - calculate the distance between William and the bucket

Thus, William has to throw the ball \(5\) units away
Example 2

 

 

If point P \((3, 4)\) and point Q \((5, 7-a)\) lie on a line parallel to the x-axis, then what is the value of \(a\)?

Solution

We know that both the points P and Q lie on a line parallel to the x-axis, thus, their y-coordinates are the same.

\[\begin{align}
7 - a &= 4\\[0.2cm]
a &= 7 - 4\\[0.2cm]
a &= 3 \end{align}\]

\(\therefore\) \(a=3\)
Example 3

 

 

Jacob and Ethan want to make a frame using the coordinates \((1, 2), (3, 2), (3, 0), (1, 0)\).

Based on the coordinates, Jacob says that the frame will be a square while Ethan says that the frame will be a parallelogram.

Can you identify who is right?

Solution

We need to draw the above coordinates on a cartesian plane to check the shape they will form.

Square on a Cartesian plane

We can clearly see that the figure thus obtained is a square as all the four sides are equal and all the four interior angles are \(90^{\circ}\).

\(\therefore\) Jacob is right.
Example 4

 

 

The cartesian coordinate plane below represents a city map with 5 different locations.

Find the cartesian coordinates for each of these locations.

A - School; B - Hospital; C - Cinema hall; D - Police department; E - Zoo

5 Places on a cartesian plane

Solution

School is situated at (1, 6)

Hospital is Situated at (3, 3)

Cinema Hall is situated at (-5, -4)

Police department is situated at (-2, 2)

Zoo is situated at (3, -4)

Example 5


 

If the four quadrants represent the following 4 states of the US:

1st Quadrant California
2nd Quadrant Florida
3rd Quadrant Texas
4th Quadrant Arizona

Can you identify in which state these points lie?
A (4, -2)
B (-3, -5)
C (1, 2)
D (-7, 1)
E (-2, -6)

Solution

Points on a cartesian plane

Point A lies in Arizona

Point B lies in Texas

Point C lies in California

Point D lies in Florida

Point E lies in Texas

Example 6

 

 

In which quadrants do the points P, Q, R and S lie in?

Points on cartesian axes

Solution

If any of the points lie on the axes or if any of the coordinate value is zero, this is how we can represent those points:

Point P lies on the positive y-axis

Point Q lies on the positive x-axis

Point R lies on the negative y-axis

Point S lies on the negative x-axis


Interactive Questions on Cartesian Coordinates

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about cartesian coordinates with the simulations and practice questions. Now you will be able to easily solve problems on cartesian coordinate plane, coordinate plane quadrants, cartesian coordinate graph, and cartesian coordinate system.

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Frequently Asked Questions (FAQs)

1. What is the difference between polar and cartesian coordinates?

When we write the cartesian coordinates of a point, we consider the distance of the point on the x-axis and y-axis.

If we consider polar coordinates of a point, then we consider the distance of the point from the origin and the angle which it forms along the x-axis.

Cartesian coordinate of a point P can be \((x, y)\), and the polar coordinates of the same point can be \((r cos\theta, rsin\theta) \), where \(r\) is the distance of the point from the origin and \(\theta\) is the angle formed by the line joining the origin and the point from the x- axis.

2. What are XY Coordinates?

x-coordinate of a point is the horizontal distance from the origin to the point and y-coordinate is the vertical distance from the origin to the point.

3. How to plot a point in the ccp?

To plot a point on the cartesian coordinate plane, we have to first check the x-coordinate of the point and draw a perpendicular line to x-axis on that value.

Next we will check the y-coordinate of the point and draw a perpendicular line to the y-axis on that value.

The point where these two lines intersect is the required point.

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