Mathematics Coordinate Geometry Coordinate System - Cartesian and Polar Coordinates

# Coordinate System – Cartesian and Polar Coordinates

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## Coordinate System – Definition

Coordinate system in Geometry is a system of one or more numbers, which uniquely determine the position (left-right, top-bottom, up-down) of the points or other geometric elements on a manifold such as Euclidean Space. -which is two dimensional, three dimensional or n-dimensional space. There are numbers of coordinate system commonly used in geometry. Number line, Cartesian, Polar, Cylindrical etc. In this article we will discuss about Cartesian and Polar Coordinate System and their relationship.[mathjax]

## Cartesian Coordinate System

Cartesian Coordinates are rectilinear two-or-three-dimensional coordinates. For two dimension, it specifies each point uniquely in a plane by a pair of numerical coordinates, $(x,y)$. Cartesian coordinate is also familiar as rectangular coordinates, because it is like moving in a box. René Descartes (1596-1650), a French philosopher, mathematician, and scientist. He shows a very strong connection between Geometry and Algebra.

let an algebra: $y = 2x-1\quad\text{…eqn(1)}$. Here for every arbitrary $x$ value, we find an associates $y$ fron the equation.

 $x$ $y$ $x=-2$ $y=2.(-2)-1=-5$ $x=-1$ $y=2.(-1)-1=-3$ $x=0$ $y=-1$ $x=1$ $y=2.1-1=1$ $x=2$ $y=2.2-1=3$

$x$ is an independent variable and $y$ here a dependent variable. Because we can choose the value of $x$ independently. and value of $y$ depends upon $x$. Graph below visualize the relationship between $x$ and $y$ for the Algebra in $eqn(1)$. Two perpendicular $X$ and $Y$ axes intersect each other on a point $(0, 0)$, therefore, the point is the origin of two axes. For every possible $x$, the pair of numerical coordinates, $(x, y)$ creates a straight line on the plane. The line is clearly a relationship between Geometry and Algebra, because every point on the straight line is a solution for $eqn(1)$. René Descartes first shows this relationship, between Algebra and Geometry. Therefore the coordinates that specify these points on the straight line is called Cartesian Coordinates, and the equation is called linear equation.

## Polar Coordinate System

Polar Coordinate is a two dimensional coordinate system, in which each point on the plane is specified by a radical distance $r$ from the reference point $0$, and an angular rotation $θ$ from the reference axis $X$. Direction of $X$ is called reference direction.

The reference point $0$, analogous to the origin of a Cartesian coordinate system, which is the pole. The straight line from the pole in the reference direction is the polar axis. distance from the pole is radius $(r)$, angle from reference direction is polar angle $(θ)$. Next we will see the relationship between these two Coordinate Systems and their conversion.

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