Cartesian Coordinate System

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as a signed distances from the origin.

Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as a signed distances from the origin.

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, one can specify a point in a space of any dimension n by use of n Cartesian coordinates, the signed distances from n mutually perpendicular hyperplanes.

The discovery of the Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x^2 + y^2 = 22.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A familiar example is the concept of the graph of a function.

Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry-related data processing.

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.



Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is x^2 + y^2 = r2.

Number line

Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—means choosing a point O of the line (the origin), a unit of length, and an orientation for the line. The latter means choosing which of the two half-lines determined by O is the positive, and which is negative; we then say that the line is oriented (or points) from the negative half towards the positive half. Then each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.

A line with a chosen Cartesian system is called a number line. Every real number, whether integer, rational, or irrational, has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum which includes the real numbers.

Cartesian coordinates in two dimensions

The modern Cartesian coordinate system in two dimensions (also called a rectangular 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. (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.) The lines are commonly referred to as the x and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For a given point P, a line is drawn through P perpendicular to the x-axis to meet it at X and second line is drawn through P perpendicular to the y-axis to meet it at Y. The coordinates of P are then X and Y interpreted as numbers x and y on the corresponding number lines. The coordinates are written as an ordered pair (x, y).

The point where the axes meet is the common origin of the two number lines and is simply called the origin. It is often labeled O and if so then the axes are called Ox and Oy. A plane with x and y-axes defined is often referred to as the Cartesian plane or xy plane. The value of x is called the x-coordinate or abscissa and the value of y is called the y-coordinate or ordinate.

The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.

Quadrants and octants

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are I (+,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("northeast") quadrant.

Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The octant where all three coordinates are positive is sometimes called the first octant; however, there is no established nomenclature for the other octants. The n-dimensional generalization of the quadrant and octant is the orthant.

The four quadrants of a Cartesian coordinate system.

The Cartesian plane consists of two directed lines that perpendicularly intersect their respective zero points.



The horizontal directed line is called the x-axis and the vertical directed line is called the y-axis. The point of intersection of the x-axis and the y-axis is called the origin and is denoted by the letter O.

The Coordinates

The position of any point on the Cartesian plane is described by using two numbers: (x, y). The first number, x, is the horizontal position of the point from the origin. It is called the x-coordinate. The second number, y, is the vertical position of the point from the origin. It is called the y-coordinate. Since a specific order is used to represent the coordinates, they are called ordered pairs.

For example, the ordered pair (5, 8) represents a point 5 units to the right of the origin in the direction of the x-axis and 8 units above the origin in the direction of the y-axis as shown in the diagram below.

We say that:

The x-coordinate of point P is 5; and the y-coordinate of point P is 8.

Or simply, we can say that:

The coordinates of point P are (5, 8).



Note the following:

For the point P(5, 8), the ordered pair is (5, 8). So:

5 is the x-coordinate, and
8 is the y-coordinate.
P(5, 8) means P is 5 units to the right of and 8 units above the origin.

Example 1

State the coordinates of each of the points shown on the Cartesian plane:


Solution:

A is 3 units to the right of and 2 units above the origin. So, point A is (3, 2).

B is 5 units to the right of and 5 units above the origin. So, point B is (5, 5).
C is 7 units to the right of and 8 units above the origin. So, point C is (7, 8).
D is 6 units to the left of and 4 units above the origin. So, point D is (–6, 4).
E is 3 units to the left of and 7 units above the origin. So, point E is (–3, 7).
F is 4 units to the left of and 6 units below the origin. So, point F is (–4, –6).
G is 8 units to the left of and 8 units below the origin. So, point G is (–8, –8).
P is 9 units to the right of and 9 units below the origin. So, point P is (9, –9).
Q is 6 units to the right of and 5 units below the origin. So, point Q is (6, –5).


Reference:
http://en.wikipedia.org/wiki/Cartesian_coordinate_system

http://www.mathsteacher.com.au/year8/ch15_graphs/01_cartesian/plane.htm