THE CARTESIAN COORDINATE SYSTEM

Cartesian coordinates allow one to specify the location of a point in the plane, or in three-dimensional space. The Cartesian coordinates or rectangular coordinates system of a point are a pair of numbers (in two-dimensions) or a triplet of numbers (in three-dimensions) that specified signed distances from the coordinate axis.  First we must understand a coordinate system to define our directions and relative position.  A system used to define points in space by establishing directions(axis) and a reference position(origin).  A coordinate system can be rectangular or polar.

Just as points on the line can be placed in one to one correspondence with the real number line, so points in plane can be placed in one to one correspondence with pairs of real number  line by using two coordinate lines.  To do this, we construct two perpendicular coordinate line that intersect at their origins; for convenience.  Assign a set of equally space graduations to the x and y axes starting at the origin and going in both directions, left and right (x axis) and up and down (y axis) point along each axis may be established.  We make one of the number lines vertical with its positive direction upward and negative direction downward.  The other number lines horizontal with its positive direction to the right and negative direction to the left.  The two number lines are called coordinate axes; the horizontal line is the x axis, the vertical line is the y axis, and the coordinate axes together form the Cartesian coordinate system or a rectangular coordinate system.  The point of intersection of the coordinate axes is denoted by O and is the origin of the coordinate system. See Figure 1.

It is basically, Two Real Number Lines Put Together, one going left-right, and the other going up-down.  The horizontal line is called x-axis and the vertical line is called y-axis.

THE ORIGIN

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

REAL NUMBER LINE

The basis of this system is the real number line marked at equal intervals. The axis is labeled (X, Y or Z). One point on the line is designated as the Origin. Numbers on one side of the line are marked as positive and those to the other side marked negative. See Figure 2.

CARTESIAN COORDINATES OF THE PLANE

A plane in which a rectangular coordinate system has been introduced is a coordinate plane or an x-y-plane. We will now show how to establish a one to one correspondence between points in a coordinate plane and pairs of real number.  If A is a point in a coordinate plane, then we draw two lines through A, one perpendicular to the x-axis and one perpendicular to the y-axis.  If the first line intersects the x-axis at the point with coordinate x and the second line intersects  the y-axis at the point with coordinate y, then we associate the pair (x,y) with the A( See Figure 2).  The number a is the x-coordinate or abscissa of P and the number b is the y-coordinate or ordinate of p; we say that A is the point with coordinate (x,y) and denote the point by A(x,y).  The point (0,0) is given the special name “The Origin”, and is sometimes given the letter “O”.

Abscissa and Ordinate:

The words “Abscissa” and “Ordinate” are words for the x and y values:

Negative Values of X and Y:

The Real Number Line, you can also have negative values.

Negative: start at zero and head in the opposite direction; See Figure 4

So, for a negative number:

For example (-3,5) means:
go left along the x axis 3 then go up 5 in the y-axis. (Quadrant II x is negative ,y is positive)

And (-3,-5) means:
go left along the x axis 3 then go down 5 in the y-axis. (Quadrant III x is negative ,y is negative)

Using Cartesian Coordinates, mark a point on a graph by how far along and how far up it is; See figure 5. The point (12,5) is 12 units along the x-axis, and 5 units up on the y-axis.

It is basically, a set of two Real Number lines.

Axis: The reference line from which distances are measured.

Example:

Go along the x direction 6  units then go up  4 units up in the y direction then “plot the dot”.

And you can remember which axis is which by:

The coordinates are always written in a certain order:

Ordered pair:

The numbers are separated by a comma, and parentheses are put around the whole thing like this:  (7,4)

Example: (7,4) means 7 units to the right (x-axis), and 4 units up (y-axis)

 

Four Quadrants: 

The quadrants are the areas on each side of the axes.  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)

Example: The point “A” (3,2) is 3 units along the x-axis, and 2 units up the y-axis.

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

Example: The point “C” (-2,-1) is 2 units along the x-axis in the negative direction, and 1 unit down the y-axis in the negative direction.

Both x and y are negative, so that point is in “Quadrant III”.

 

Here’s another example.

 

The points may now be defined in a three dimensional volume of space. This permits to define points in three dimensions from the origin.  The Cartesian coordinates (x,y,z) of a point in three-dimensions specify the signed distance from the origin along the x, y, and z-axes, respectively.  Z-axis points become the third entry when defining coordinate locations.

Given the above corner-of-room analogy, we could form the Cartesian coordinates of the point at the top of your head, as follows. Imagine that you are five meters tall  the z-axis, and that you walk two meters from the origin along the x-axis, then turn left and walk parallel to the y-axis four meters into the room. The Cartesian coordinates of the point at the top of your head would be (2,4,5).

For example, a notation of (2,4,5) corresponds to the value of X2, Y4, and Z5. See Figure 8.

 

3 DIMENSIONS

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

DIMENSIONS: 1, 2, 3 AND MORE …

• left-right
• so any position needs just one number

• left-right, and
• up-down
• so any position needs two numbers

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