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# Rectangular Coordinate System

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 Sub Topics Rectangular co-ordinate system is also called as a cartesian coordinate system. It is a coordinate system that defines every point differently in a plane by the pair of numerical coordinates. Rectangular coordinate system was invented by Rene Descartes in 17th century. Using this system, the geometric shapes are described by the Cartesian equation. Cartesian coordinates are the foundation for analytic geometry and they influence the geometric interpretations for other branches of mathematics like linear algebra, complex analysis, calculus, group theory and more.

## Definition of Rectangular Coordinate System

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A rectangular coordinate system is formed by two number lines, the horizontal and the vertical, which intersects at the common point. The point of intersection is called as ''origin''. The two axes divide the plane into 4 regions called as ''quadrants'' and they are numbered from I to IV as shown in the figure below:

Each point on the plane are defined by the pair of numbers called as ordered pair. The ordered pair which is measured on horizontal distances is called as the ''Abscissa" and the ordered pair which is measured on vertical distances is called as the ''Ordinate''. The number in the ordered pair which are connected with the point are called as the coordinate of the point.
Abscissa is called as the first coordinate or x- coordinate of the ordered pair, and the ordinate is called as second coordinate or y-coordinate.

## Rectangular Coordinate System in Two Dimensions

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The two dimensional rectangular coordinates are the one which are defined by 2 axes, forming a xy plane having right angles to each other. The horizontal axis is defined as x, that is the abscissa and the vertical axis as y that is the ordinate. The point of intersection is called as origin.

To define a specific point on a two dimensional coordinate system, we denote the x unit first and then the y unit in the form of (x, y). The axes intersecting at the origin forming 4 quadrants which are numerated in a clockwise direction by the roman numbers as shown in the figure below:

The distance between a point P and the origin O is calculated using the Pythagorean theorem

$OP_1$ = $\sqrt{x_1^2+y_1^2}$

$\therefore$ The distance between P$_1$ and P$_2$ is

$d$ = $P_1P_2$ = $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

## Rectangular Coordinate System in Three Dimensions

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The three dimensional rectangular coordinates are the one which has 3 axes, located at the right angles to each other. And, they form a three dimensional space. Here, the three axes are defined by x (abscissa), y (ordinate) and z, which is called as applicate. Even here, the point where the axes intersect is called as origin, denoted by O. The particular values on the axes are denoted in the form of (x, y, z).

Using the Pythagorean theorem, the distances between a point P and the origin O is calculated and we get,

$OP_1$ = $\sqrt{x_1^2+y_1^2+z_1^2}$

The shortest distance left P$_1$ and P$_2$ is also calculated in the same way

$d$ = $P_1P_2$ = $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

## How to Find the Rectangular Coordinates?

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Given below are the few example problems based on rectangular coordinate which will help you to get an idea on how to find the rectangular coordinates.

### Solved Examples

Question 1: Find the ordered pair solution of $y$ = $\frac{x}{x-4}$, that corresponds to x = 6.
Solution:
Given:

$y$ = $\frac{x}{x-4}$

$y$ = $\frac{6}{6-4}$

y = 3

$\therefore$ the ordered pair solution is (6, 3).

Question 2: Find the ordered pair solution of $y$ = $\frac{3x}{x+3}$, that corresponds to x = -2.
Solution:
Given:

$y$ = $\frac{3x}{x+3}$

$y$= $\frac{3(-2)}{-2+3}$

y = -6

$\therefore$ the ordered pair solution is (-2, -6).

Question 3: Construct a graph for the ordered pairs solution of y = x2 - 2x, when x = -1, 0, 1 and 2.
Solution:
Given:

y = x2 - 2x

When x = -1
y = 1 + 2
y = 3.

When x = 0
y = 0

When x = 1
y = -1

When x = 2
y = 0

$\therefore$ the ordered pairs are (-1, 3), (0, 0), (1, -1), and (2, 0).