Cartesian Coordinates (2024)

Cartesian coordinates can be used to pinpoint where we are on a map or graph.

Cartesian Coordinates

Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is:

The point (12,5) is 12 units along, and 5 units up.

They are also called Rectangular Coordinates because it is like we are forming a rectangle.

X and Y Axis

	The left-right (horizontal) direction is commonly called X.
	The up-down (vertical) direction is commonly called Y.

Put them together on a graph ...

... and we are ready to go

Where they cross over is the "0" point,
we measure everything from there.

The X Axis runs horizontally through zero
The Y Axis runs vertically through zero

Axis: The reference line from which distances are measured.

The plural of Axis is Axes, and is pronounced ax-eez

Example:

Point (6,4) is

6 units across (in the x direction), and

4 units up (in the y direction)

So (6,4) means:

Play With It !

Now is a good time to play with

Interactive Cartesian Coordinates

to see for yourself how it all works.

Like 2 Number Lines Put Together

It is like we put two Number Lines together, one going left-right, and the other going down-up.

Direction

	As x increases, the point moves further right. When x decreases, the point moves further to the left.
	As y increases, the point moves further up. When y decreases, the point moves further down.

Writing Coordinates

The coordinates are always written in a certain order:

the horizontal distance first,
then the vertical distance.

This is called an "ordered pair" (a pair of numbers in a special order)

And usually the numbers are separated by a comma, and parentheses are put around the whole thing like this:

(3,2)

Example: (3,2) means 3 units to the right, and 2 units up

Example: (0,5) means 0 units to the right, and 5 units up.

In other words, only 5 units up.

The Origin

The point (0,0) is given the special name "The Origin", and is sometimes given the letter "O".

Abscissa and Ordinate

You may hear the words "Abscissa" and "Ordinate" ... they are just the x and y values:

Abscissa: the horizontal ("x") value in a pair of coordinates: how far along the point is
Ordinate: the vertical ("y") value in a pair of coordinates: how far up or down the point is

"Cartesian" ... ?

They are called Cartesian because the idea was developed by the mathematician and philosopher Rene Descartes who was also known as Cartesius.

He is also famous for saying "I think, therefore I am".

What About Negative Values of X and Y?

Just like with the Number Line, we can also have negative values.

Negative: start at zero and head in the opposite direction:

Negative x goes to the left
Negative y goes down

So, for a negative number:

go backwards for x
go down for y

For example (-6,4) means:
go back along the x axis 6 then go up 4.

And (-6,-4) means:
go back along the x axis 6 then go down 4.

Four Quadrants

When we include negative values, the x and y axes divide the space up into 4 pieces:

Quadrants I, II, III and IV

(They are numbered in a counterclockwise direction)

In Quadrant I both x and y are positive, but ...

in Quadrant II x is negative (y is still positive),
in Quadrant III both x and y are negative, and
in Quadrant IV x is positive again, while y is negative.

Like this:

Quadrant	X (horizontal)	Y (vertical)	Example
I	Positive	Positive	(3,2)
II	Negative	Positive
III	Negative	Negative	(-2,-1)
IV	Positive	Negative

Example: The point "A" (3,2) is 3 units along, and 2 units up.

Both x and y are positive, so that point is in "Quadrant I"

Example: The point "C" (-2,-1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction).

Both x and y are negative, so that point is in "Quadrant III"

Note: The word Quadrant comes form quad meaning four. For example, four babies born at one birth are called quadruplets, a four-legged animal is a quadruped. and a quadrilateral is a four-sided polygon.

Dimensions: 1, 2, 3 and more ...

Think about this:

1	The Number Line can only go: left-right so any position needs just one number
2	Cartesian coordinates can go: left-right, and up-down so any position needs two numbers
3	How do we locate a spot in the real world (such as the tip of your nose)? We need to know: left-right, up-down, and forward-backward, that is three numbers, or 3 dimensions!

The Number Line can only go:

left-right

so any position needs just one number

Cartesian coordinates can go:

left-right, and
up-down

so any position needs two numbers

How do we locate a spot in the real world (such as the tip of your nose)? We need to know:

left-right,
up-down, and
forward-backward,

that is three numbers, or 3 dimensions!

3 Dimensions

Cartesian coordinates can be used for locating points in 3 dimensions as in this example:

Here the point (2, 4, 5) is shown in
three-dimensional Cartesian coordinates.

In fact, this idea can be continued into four dimensions and more - I just can't work out how to illustrate that for you!

753,754,755,756,1142,1143,1144, 3208, 3209, 3210

Interactive Cartesian Coordinates Hit the Coordinate Game Polar and Cartesian Coordinates The Number Line Graphs Index

FAQs

What is the formula for Cartesian coordinates? ›

The cartesian form of a plane can be represented as ax + by + cz = d where a, b, and c are direction cosines that are normal to the plane and d is the distance from the origin to the plane.

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How to write Cartesian coordinates? ›

The Cartesian , or rectangular , coordinate system consists of a horizontal x-axis and a vertical y-axis. The point where the axes cross is called the origin . Any point can be described as the distance it is from the origin along the x-axis and along the y-axis and is written as (x, y).

Find Out More ›

How do you solve Cartesian equations? ›

A cartesian equation of a curve is simply finding the single equation of this curve in a standard form where xs and ys are the only variables. To find this equation, you need to solve the parametric equations simultaneously: If y = 4t, then divide both sides by 4 to find (1/4)y = t.

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How do you calculate Cartesian form? ›

A point, a line, or a plane can be easily represented in a three-dimensional plane, across the x-axis, y-axis, z-axis, in cartesian form. The cartesian form of representation of a point is (x, y, z), the line is (x - x₁)/a = (y - y₁)/b = (z - z₁)/c, and the plane is ax + by + cz = d.

Get More Info Here ›

What are Cartesian coordinates simple? ›

A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis.

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How do you convert coordinates to Cartesian coordinates? ›

Summary. To convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) : x = r × cos( θ ) y = r × sin( θ )

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How do you teach Cartesian Coordinates? ›

Talk about the Cartesian plane quadrants, having students label them "I, II, III and IV." Point out which directions are positive (right on the x-axis and up on the y-axis) and which directions are negative (left on the x-axis and down on the y-axis), and have students mark out scales on each axis.

Read The Full Story ›

What are the XY Cartesian Coordinates? ›

In the Cartesian coordinate system, the x and y coordinates are part of the x-axis and y-axis in a 2D space. For a point in space, the x and y coordinates are written as an ordered pair, (x, y). The first number represents the point's position on the x-axis, and the second number represents its position on the y-axis.

Keep Reading ›

How do you reveal coordinates? ›

Get the coordinates of a place

On your computer, open Google Maps.
Right-click the place or area on the map. This will open a pop-up window. You can find your latitude and longitude in decimal format at the top.
To copy the coordinates automatically, left click on the latitude and longitude.

How do you find the point of a Cartesian equation? ›

Ans : The cartesian equation of a point, line or plane defines the equation of the figure in the x, y, and z-axis. It provides the equation along with its direction. Where l, m, and n are the direction ratios. Ans : The standard equation form of a line is y=mx+c.

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How do you locate points on the Cartesian plane? ›

The horizontal axis is called the x-axis and the vertical one is the y-axis. These axes intersect with each other at the origin whose location is given as (0, 0). Any point on the cartesian plane is represented in the form of (x, y).

Cartesian Coordinates (2024)