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:
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| 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
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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:
Go along 6 and then go up 4 then "plot the dot".
And you can remember which axis is which by:
x is A CROSS, so x is ACROSS the page.
| | 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.
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| Four QuadrantsWhen 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: so any position needs just one number |
| 2 | Cartesian coordinates can go: 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! |
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
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.
How do you find the Cartesian coordinates of a point? ›
Cartesian coordinates (x, y) is related with its polar coordinates (r, α) by the relation; x = r cosα , y = r sinα . Given here polar coordinates as ; (5, π−arctan(4/3)) ==> r = 5 & α = π−arctan(4/3) so that x = 5 cosα , y = 5sinα .
How to write Cartesian coordinates? ›
The Cartesian coordinates of a point in the plane are written as (x,y). The first number x is called the x-coordinate (or x-component), as it is the signed distance from the origin in the direction along the x-axis.
What do Cartesian coordinates look like? ›
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).
How to 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.
How to plot Cartesian coordinates? ›
Step 1: Identify the point in question, (x,y). Step 2: Determine its x-coordinate, which will be the horizontal distance from the origin. Step 3: Determine its y-coordinate, which will be the vertical distance from the origin. Step 4: Plot or label the point (depending on what the question is asking).
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.
How do you calculate Cartesian form? ›
To express in Cartesian form, translate a vector, complex number or function from its current form (usually polar or spherical) into x, y, and z coordinates. For example, a complex number r(cos θ +i sin θ) converts to x + yi, where x=r cos θ and y=r sin θ.
What is the order of Cartesian coordinates? ›
The first and second coordinates are called the abscissa and the ordinate of P, respectively; and the point where the axes meet is called the origin of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in (3, −10.5).
How to solve a Cartesian coordinate plane? ›
The Cartesian coordinate plane allows us to visualize ordered pairs by representing the inputs along horizontal number line called the x axis and outputs along a vertical number line called the y axis. Move the number of units in the x component horizontally. If x is positive, move right. If x is negative, move left.
The Cartesian plane, also known as the coordinate plane, has four quadrants that are labeled counterclockwise. The value of the origin on the -axis is zero. If you think of the -axis as a number line, the numbers to the right of zero are positive values, and those to the left of zero are negative values.
How do you find the Cartesian equation of a point? ›
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.
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.
What is Cartesian coordinate points? ›
A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the axes) that go through a common point (the origin), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes.
How to solve points on a Cartesian plane? ›
Given a point, (a, b), you plot the point in the Cartesian plane by finding the number a on the horizontal, or x-, axis; then you trace directly up or down from that spot, moving parallel to the vertical, or y-, axis, until you reach the height of the number b. This spot is the location of the point (a, b).
