Rectangular Cartesian Coordinate System :: Ordered Pair, Cartesian Product, Coordinate Plane, Quadrants

Learning Analytical/Coordinate/Cartesian Geometry
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Ordered Pair

It is represented as (First element, Second element)

Eg:	(x, y), (p, q), (3, -5),

The order pair (x, y) contains two elements ‘x’ and ‘y’. They are arranged in such a way that ‘x’ is the first element and ‘y’ the second element.

Set of Ordered Pairs

A set of ordered pairs is represented as {[x], (x, y)}

Eg:	Consider the following data relating to number of pens (x) and their cost (y) Number of pens (x)   1     2     3     4     5     6   Cost of Pens (y)  15  25  35  45  55  65

Since the two variables are related, the data can be represented as ordered pairs which indicate the relationship between the two variables.

The data written as a "Set of Ordered Pairs" would be
      {(1, 15), (2, 25), (3, 35), (4, 45), (5, 55), (6, 65)}
The first element of the set represents ‘x’ i.e. the number of pens and the second element represents ‘y’ i.e. the cost of pens. The numbers in all the elements of the sets (i.e. in all the brackets) are written in the same order. (x, y)

The ordered pair (3, 35)
      ⇒ The value of ‘y’ is 35 when x is 3. (Cost is 35 when the number of pens is 3)

Order of writing elements is important.

(3, 35) ⇒ The value of ‘y’ is 35 when x is 3.
(35, 3) ⇒ The value of ‘y’ is 3 when x is 35.

First Coordinate

Second Coordinate

Cartesian Product of two sets

It is denoted with the symbol ‘X’ between the names of the two sets.

Eg:

For two sets A and B, the set of all possible ordered pairs (x, y) with x ∈ A and y ∈ B is called the Cartesian Product of A and B. It is represented as A X B [Read as A cross B] A X B = { (x, y)/ x ∈ A and y ∈ B} Where A = {1, 2, x} and B = (1, q, 5}; A X B = { (1, 1), (1, q), (1, 5), (2, 1), (2, q), (2, 5), (x, 1), (x, q), (x, 5)}

Number of Ordered Pairs

For two sets A and B, n (A X B) = n (A) × n (B)

Eg:	Where A = {1, 2, x} and B = (1, q, 5}; n (A) = 3 and n(B) = 3. Therefore, Number of elements in the Cartesian Set A X B ⇒ n (A x B) = n (A) × n (B) = 3 × 3 = 9

Cartesian Product of the real number set

R = (− ∞ , + ∞)

Where "x ∈ R₁" and "y ∈ R₂"

R X R = {(x₁, y₁), (x₂, y₂), (),…… }

Number of elements in R X R
n (R) = ∞ ⇒ n (R₁) = n (R₂) = ∞
Therefore, Number of elements in R x R

⇒ n (R X R)	=	n (R1) × n (R2)
	=	∞ × ∞
	=	∞

Plane

Naming the Plane

It can also be named using any three points on it. Plane AHM or Plane APM etc.

“Descartes”, the French mathematician, initiated two famous discoveries which unified algebra and geometry 1) Any ordered pair of numbers can be represented by a point in a plane. 2) Any curve in a two dimensional space can be defined by an equation in x and y

Association of "Points in a Plane & Cartesian Product of the Real Number Set"

If λ (x, y) represents the set of points making up the plane,
Then a one – one function from λ (x, y) to R X R can be identified.
⇒ f : λ (x, y) → R X R [f is a one – one function]

For every element in λ (x, y) (i.e. for every point in the plane) a unique element can be identified in the set R X R (i.e. a unique ordered pair of real numbers can be related).

f is called a coordinate system on λ (x, y).

Every plane has a coordinate system.

Coordinate System on a Plane

Parameters

• Origin
  O being called the Origin of the Plane. It corresponds to the Ordered Pair O (0, 0) on the plane.

• Coordinate Axes
  x’x and y’y are called the coordinate axes. Each of the coordinate axes is a Coordinate Line. The coordinate lines are mutually perpendicular. The Origins of the Coordinate Lines Coincides with the Origin of the plane.
  • x - axis:
    The horizontal coordinate line is named x – axis and is denoted as x’x or x’ o x
  • y - axis:
    The vertical coordinate line is named y – axis and is denoted as yy’ or yoy’

The Origin and the coordinate axes together define the nature of the coordinate system formed by the coordinate plane.

Constructing the Coordinate Plane

• Taking an arbitrary Point ‘O’ in a plane and placing two coordinate lines having the same unit length, one (x’x) horizontally and the other (yy’) vertically, with their origins coinciding with the point ‘O’.
  (Or)
• Drawing a horizontal number line anywhere on the plane with numbers marked with a unit length and then drawing a vertical number line (with the same unit length) perpendicular to the horizontal line such that the Zero (0) of the both the lines coincide.

Coordinates of a Point

"x,y" are called the coordinates of “P” with respect to coordinate system f on the plane α.

Written as P (x, y) [P with coordinates x and y]

Eg:	(1)	P (3,1) ⇒ 3 and 1 are the coordinates of P [Read as P with coordinates 3 and 1] "3" is also called the "First coordinate" or "Abscissa" or "Argument" "1" is also called the "Second coordinate" or "Ordinate" or "Entry"
	(2)	C (−4,3) ⇒ − 4 and 3 are the coordinates of C [Read as C with coordinates − 4 and 3] "−4" is also called the "First coordinate" or "Abscissa" or "Argument" "3" is also called the "Second coordinate" or "Ordinate" or "Entry"

The Four Quadrants

• First Quadrant
  This is bounded by the two rays "OX" and "OY". It is denoted as "Q₁" or as "XOY"

  The horizontal and vertical axes have positive numbers over them.

  The first coordinate as well as the second coordinate of all the points in this Quadrant are positive.

• Second Quadrant
  This is bounded by the two rays "OX'" and "OY". It is denoted as "Q₂" or as "X'OY"

  The horizontal axis has negative numbers over it and the vertical axis has positive numbers over it.

  The first coordinate is negative and the second coordinate is positive for all the points in this Quadrant.

• Third Quadrant
  This is bounded by the two rays "OX'" and "OY'". It is denoted as "Q₃" or as "X'OY'"

  The horizontal and vertical axes have negative numbers over them.

  The first coordinate as well as the second coordinate of all the points in this Quadrant are negative.

• Fourth Quadrant
  This is bounded by the two rays "OX" and "OY'". It is denoted as "Q₄" or as "XOY'"

  The horizontal axis has positive numbers over it and the vertical axis has negative numbers over it.

  The first coordinate is positive and the second coordinate is negative for all the points in this Quadrant.

Location of Positive and Negative numbers on the Coordinate lines forming the Coordinate plane

• Positive integers lie:
  • On the horizontal line – Towards the right of the origin
  • On the vertical line – Towards the north of origin (upwards)
• Negative integers lie:
  • On the horizontal line – Towards the left of the origin
  • On the vertical line – Towards the south of the origin (down wards)

Positive and Negative Sides in the Coordinate Plane

• With respect to the x – axis:
  • The upward direction (towards north) is +ve
  • The downward direction (towards south) is –ve
• With respect to the y – axis
  • The right hand side (towards east) is +ve
  • The left hand side (towards west) is –ve

Author Credit: The Edifier

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