Cartesian Coordinates

Background |  Ordered Pairs  | The Cartesian Coordinate System in 2-D  |  The Cartesian Coordinate System in 3-D

Background

We assume that you are familiar with the representation of a number on a number line.

Ordered Pairs

An ordered pair is simply a pair of elements written in a certain order (order is important). For example, the numbers 5 and 2 can form two ordered pairs:

(5,2) and (2,5).

In general, given two numbers x and y, the two ordered pairs (x,y) and (y,x) are different, unless x = y.

The number x is called the first component and y the second component of the ordered pair (x,y).

Ordered pairs have many uses. It can be used to represent an object with two distinguishable parts. For example, a complex number x+iy can be represented by the ordered pair (x,y). In an Excel spreadsheet we use a letter and a number to denote a cell on the sheet.

On a street map we use the same method to locate a street.

Cartesian Coordinate System in 2-D

Ordered pairs are useful in representing relationships between two sets of data, in the form of a graph. These can then be represented by points on a grid known as the Cartesian Coordinate System in 2-D.

The following example illustrates how to represent an ordered pair as a point on this grid.

To plot the point represented by (2,4):

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If a point P is represented by (a,b) we call a and b the coordinates of P. In 2-D Cartesian Coordinates, the first coordinate in the ordered pair is called the x-coordinate and the second coordinate in the ordered pair is called the y-coordinate.

Exercise 1.

Enter the correct coordinates for the points plotted below, A-L.
Remember to separate your x and y coordinates by a comma.

If you have trouble with a point, place your mouse cursor over a point to see its coordinates.

A: ( ) B: ( ) C: ( ) D: ( ) E: ( ) F: ( )

G: ( ) H: ( )  I: ( ) J: ( ) K: ( ) L: ( )

 

 

 

 

 

 

 

The two axes partition the plane into 4 regions, called quadrants.
They are referred to as the 1^st, 2^nd, 3^rd, and 4^th quadrant as shown.

• If the x and y coordinates are both positive, the point is located in the 1st quadrant.
• If the x coordinate is negative and the y coordinate is positive, the point is located in the 2nd quadrant.
• If the x and y coordinates are both negative, the point is located in the 3rd quadrant.
• If the x coordinate is positive and the y coordinate is negative, the point is located in the 4th quadrant.

Example 2.

Look at the following example to see which quadrants various points are located in. Check the grid below to confirm the solution.

Cartesian Coordinate System in 3-D

An ordered triple is a list of 3 elements written in a certain order. As with ordered pairs, order is important.

For example, the numbers 1, 2 and 3 can form 6 ordered triples: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).

An ordered triple such as (x,y,z) can be represented by a point in a coordinate system with three axes, the Cartesian Coordinate System in 3-D. To draw the three axes, first draw a 2-D Cartesian coordinate system and label the axes y and z. Next, we draw a third axes that it is perpendicular to the other two and label the axes x (imagine that the x-axis is pointing directly at you).

The intersection of the three axes is called the origin. It represents the triple (0,0,0) and will be denoted by 0.

The following example illustrates how to represent an ordered triple as a point on this 3-dimensional grid.

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Background |  Ordered Pairs  | The Cartesian Coordinate System in 2-D  |  The Cartesian Coordinate System in 3-D