Geometry - Ordered Pair
- What Is an Ordered Pair?
- Ordered Pair Definition
- Cartesian Coordinate System for Ordered Pair
- Identifying Points with Ordered Pairs
- Identifying the Quadrants for the Ordered Pair
- Cartesian Product for Ordered Pairs
- Equality Property of Ordered Pairs
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What Is an Ordered Pair?
An ordered pair is a concept in mathematics that is used to represent a point or coordinate in two-dimensional space. The syntax for writing an ordered pair is to write it in open brackets or parenthesis, separated by a comma.
For example, `(x,y)` represents the coordinate of the `x`-axis in the horizontal direction, while the second number `y`, represents the coordinate of the `y`-axis in the vertical direction.
Consider an example `(4,5)`. In this example, `4` represents the `x`-coordinate and `5` represents the `y`-coordinate. This is an ordered pair, which denotes that the point `(4,5)` lies in a two-dimensional space.
The important point to note in this example is that `(4,5)` will not be equal to `(5,4)`. As in `(5,4), 5` represents the `x`-coordinate and `4` represents the `y`-coordinate.
Ordered Pair Definition
An ordered pair can be defined as a fixed order of numbers used to represent a point in two-dimensional space.
Examples of an ordered pair are `(2,5), (-5,6), (3,9), (8,2)`, etc.
Cartesian Coordinate System for Ordered Pair
This system is named after the French mathematician and philosopher René Descartes. It is a system of two-dimensional space used in mathematics and science to locate and denote points. It uses horizontal and vertical axes to locate the positions of objects with a reference called the origin. The horizontal axis is called the `x`-axis while the vertical axis is called the `y`-axis. A cartesian system is shown below.
From the above figure, the following observations can be made:
- A point marked origin is represented with `(0,0)`. It is also called a reference point from which all the coordinates are measured.
- The distance of the `x`-coordinate from the origin is called abscissa, while the distance of the `y`-coordinate from the origin is called ordinate.
Identifying Points with Ordered Pairs
The following steps can be used to mark the points on the cartesian plane.
- Identify the ordered pair: Look for the ordered pair `(x,y)` representing the point to be marked. The value of `x` represents abscissa on the horizontal axis (`x`-axis), and the second value `y` represents ordinate on the vertical axis (`y`-axis).
- Move horizontally: From the reference point, which is the origin, move horizontally depending on value of `x`.
- If `x` is positive, move rightward from the origin to `|x|`.
- If `x` is negative, move leftward from the origin to `|x|`.
- Move vertically: From the reference point, that is the origin, move vertically depending on the value of `y`.
- If `y` is positive, move upward from the origin to `|y|`.
- If `y` is negative, move downward from the origin to `|y|`.
Note: `|x|` is the absolute value of `x` and `|y|` is the absolute value of `y`.
- Mark the point: The point obtained after the intersection of movements is the identified ordered pair; mark the point with a capital letter.
Example `1`: Mark the point `(4,5)` on the cartesian coordinate with the ordered pair.
Solution:
Following the above steps, moving `4` units rightward and `5` units upward. The ordered pair can be identified as a point `P(4,5)`.
Example `2`: Mark the point `(-5,3)` on the cartesian coordinate with the ordered pair.
Solution:
Following the above steps, moving `5` units leftward and `3` units upward, the ordered pair can be identified as a point `Q(-5,3)`.
Identifying the Quadrants for the Ordered Pair
The cartesian coordinate system divides the space into four parts, as can be seen in the image below. These parts are called quadrants.
The names of the quadrants are known as the first, second, third, and fourth quadrants. Each quadrant is identified by the sign of the coordinate, as shown in the image above. The value of coordinates is also used to identify the quadrants.
- If `x>0`, and `y>0`, then the ordered pair will lie in Quadrant I.
- If `x<0`, and `y>0`, then the ordered pair will lie in Quadrant II.
- If `x<0`, and `y<0`, then the ordered pair will lie in Quadrant III.
- If `x>0`, and `y>0`, then the ordered pair will lie in Quadrant IV.
We can understand this from the following examples.
Example: Plot ordered pairs `A (2,4), B (-3,3), C (-4,-2)`, and `D (3,-4)` and identify the quadrants for each pair.
Solution:
The ordered pairs can be located as shown in the image below.
- `(2,4)` lies in Quadrant I.
- `(-3,3)` lies in Quadrant II.
- `(-4,-2)` lies in Quadrant III.
- `(3,-4)` lies in Quadrant IV.
Cartesian Product for Ordered Pairs
A cartesian product is a mathematical operation that is used to multiply two cartesian sets to get a third cartesian set. Suppose `A` and `B` are two sets defined for ordered pairs. Here, the first element of each ordered pair comes from set `A`, while the second element comes from set `B`.
Mathematically,
`A\times B=\{(a,b)|a\in A,b\in B\}`
Here, `A xx B` represents the cartesian product of sets `A` and `B`.
`(a,b)` is an ordered pair that `a` belongs to set `A` and `b` belongs to set `B`.
The vertical line | is read as ‘such that’ and separates the conditions for forming ordered pairs.
`a\in A` indicates that the `a` is an element in set `A`.
`b\in B` indicates that the `b` is an element in set `B`.
The new set after the operation `A xx B` contains all possible combinations of elements in sets `A` and `B`. If set `A` has `p` elements and set `B` has `q` elements, then `A xx B` will have `p xx q` elements.
Example `1`: Let set `A={3,4} and B={1,2}`. Find the cartesian product of the sets.
The Cartesian product is given as, `A xx B={(3,1),(3,2),(4,1),(4,2)}`.
Equality Property of Ordered Pairs
The equality property of ordered pairs states that two ordered pairs can be said to be equal if the corresponding elements of both ordered pairs are equal.
If `(x,y)=(p,q)`, then `x=p`, and `y=q`.
For example, `(x,y)=(3,5)`and `(3,5) = (a,b)` find `(a,b)` by the equality property.
By the equality property,
`x=3=a` and `y=5=b`.
Therefore, `(a,b)=(3,5)`.
Solved Examples
Example `1`: What will `(5,3)` denote on the cartesian coordinate?
Solution:
In an ordered pair, the terms are represented as `(x,y)`, wherein `x` represents the abscissa (`x`-direction) in the horizontal direction on the cartesian coordinate plane and `y` represents the ordinate (`y`-direction).
Here, `5` represents the movement on the `x`-axis in the positive-`x` direction, and `3` represents the movement on the `y`-axis in the positive-`y` direction.
Now, by the sign of the coordinates, which are positive, a quadrant can be identified, which is Quadrant I for this ordered pair.
Example `2`: Can you identify the quadrant for `(-1,5), (3,-5), (4,4)`, and `(-1,-3)` points on the cartesian plane?
Solution:
`(-1,5)` lies in Quadrant II.
`(3,-5)` lies in Quadrant IV.
`(4,4)` lies in Quadrant I.
`(-1,-3)` lies in Quadrant III.
Practice Problems
Q`1`. Which of the following ordered pairs lies in Quadrant II?
- `(-10.11)`
- `(4,-7)`
- `(-9,0)`
- `(4,9)`
Answer: c
Q`2`. Which is the midpoint between points `P(4,6)` and `(10,12)`?
- `(3,9)`
- `(7,9)`
- `(2.9)`
- `(-7,-6)`
Answer: b
Q`3`. If a point is denoted by `A (3x+1, y-4)` and another point is denoted by `B (10, 8)`. If `A=B`, find the coordinates of point `A`.
- `(-6,8)`
- `(2,-7)`
- `(0,0)`
- `(10,8)`
Answer: d
Q`4`. Which ordered pair represents the origin in the cartesian coordinate system?
- `(1,1)`
- `(0,0)`
- `(2,2)`
- `(4,4)`
Answer: b
Q`5`. Identify the quadrant of the point `(-5,9)`.
- `I`
- `II`
- `III`
- `IV`
Answer: b
Frequently Asked Questions
Q`1`. What are the midpoints, and how can they be calculated using ordered pairs?
Answer: A midpoint is the point that lies at the center of the line segment. By using the cartesian coordinate system a midpoint can be calculated as follows. For example, the midpoint of the points `(3,8)` and `(6,10)` is `((3+6)/2,(8+10)/2)=(9/2,9)`
Q`2`. How can ordered pairs be used in geometry?
Answer: In geometry, it is used to represent the location of the points or objects. Ordered pairs are also used to find the lengths and ratios of the line segments in the cartesian system.
Q`3`. Are there any other coordinate systems than the cartesian coordinate system?
Answer: Yes, there are other coordinate systems as well. Such as the spherical coordinate system and the cylindrical coordinate system.
Q`4`. Can we name the quadrant in the clockwise direction?
Answer: No, the internationally accepted standard notation of the coordinate system is in the clockwise direction only.