A coordinate system is used to determine each point uniquely in a plane. The study of the coordinate system led to innovations and research in the field of coordinate geometry. This study was initially developed by the French Philosopher and Mathematician Rene Descartes.
We know that a number line is a line that is used to represent the integers in both positive and negative directions from the origin. Rene Descartes invented the idea of placing two such lines perpendicular to each other on a plane and locating points on the plane by referring them to these lines. Though the two perpendicular lines can be in any direction, we usually choose one line on the vertical axis and the other perpendicular to it on the horizontal axis. In this article, we mainly discuss two types of coordinate systems.
Different Types of Coordinate Systems
There are mainly two types of coordinate systems, as listed below:
Cartesian Coordinate System
As stated above, it uses the concept of mutually perpendicular lines to denote the coordinate of a point. To locate the position of a point in a plane using two perpendicular lines, we use the Cartesian coordinate system. Points are represented in the form of coordinates (x, y) in two-dimension with respect to the x- and y- axes.
The x-coordinate of a point is its perpendicular distance from the y-axis measured along the x-axis, and it is known as abscissa. The y-coordinate of a point is its perpendicular distance from the x-axis measured along the y-axis, and it is known as ordinate. Read More.
Polar Coordinate System
Here, a point is chosen as the pole, and a ray from this point is taken as the polar axis. Basically, we have two parameters, namely angle and radius. The angle Ɵ with the polar axis has a single line through the pole measured anti-clockwise from the axis to the line.
The point will have a unique distance from the origin (r). Thus, a point in the polar coordinate system is represented as a pair of coordinates (r, Ɵ). The pole is represented by (0, Ɵ) for any value of Ɵ, where r = 0.
(r, Ɵ), (r, Ɵ + 2π) and (-r, Ɵ + π) are all polar coordinates for the same point.
The distance from the pole is called the radial coordinate, radial distance or simply radius and the angular coordinate, polar angle or azimuth.
Consider the figure below that depicts the relationship between polar and cartesian coordinates.
X = r cos Ɵ and y = r sin Ɵ
r = (x2 + y2) ½ and tan Ɵ = (y/x)
The polar equation of a curve consists of points of the form (r, Ɵ).
| In the case of a circle, the general equation for a circle with a centre at (R, β) and
radius a is r2 – 2rR cos(Ɵ – β) + R2 = a2. |
Radial lines (those running through the pole) are represented by the equation: Ɵ = β.
Cartesian Formulae for the Plane
Distance between two points
The distance between two points of the plane (x1, y1) and (x2, y2) is given by
d = [(x2 – x1)2 + (y2 – y1)2 ]1/2
In the case of a three-dimensional system, the distance formula between the points (x1, y1, z1) and (x2, y2, z2) is
d = [(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2 ]1/2
Representation of a vector
In two dimensions, the vector from the origin to the point with the Cartesian coordinates (x, y) can be written as r = xi + yj, where i = (1,0) and j = (0,1) are unit vectors in the direction of the x-axis and y-axis, respectively.
In the case of three dimensions, we will have r = xi + yj + zk, where k = (0,0,1) is the unit vector in the direction of the z-axis.
Also, Read:
Example Problems
Example 1: On graph paper, plot the point where Ajay has kept the lamp. He describes the situation as: the lamp is kept on a table which is taken as the reference. The lamp has a height of 2 units and is kept at the edge of the beginning end of the table.
Solution: As per the statement given in the question, the reference table is to be considered as the origin (0, 0). The position of the lamp, as stated above, clearly points out to be on the coordinate (0, 2). Thus, the point marked A is the position of the lamp with respect to the table reference marked as B. The following graph illustrates this.
Example 2: Find the distance between the two points (4,5,6) and (9,8,2) in the x-y plane.
Solution: d = [(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2 ]1/2
Given: (x1, y1, z1) = (4,5,6) and (x2, y2, z2) = (9, 8, 2)
Therefore, x2 – x1 = 9 – 4 = 5.
Similarly, y2 – y1 = 8 – 5 = 3 and z2 – z1 = 2 – 6 = -4
Hence, d2 = 52 + 32 + (-4)2
or, d = √50 units.
Example 3: The points
A) An equilateral triangle
B) An isosceles triangle
C) A right-angled triangle
D) None of these
Solution:
Let
Then
Similarly
Hence, it is an isosceles triangle.
Example 4: If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), then
(A) ax + by = 0
(B) ax – by = 0
(C) bx + ay = 0
(D) bx – ay = 0
Solution:
On simplification, we get bx – ay = 0
Trick: The locus will be the right bisector of the line joining the given points, therefore, the line must pass through the mid-points of the given point, i.e., (a, b). Obviously, the line given in option (d) passes through (a, b).
Polar Coordinates
Frequently Asked Questions
How do we represent a point in a Cartesian coordinate system?
A point can be represented as (x, y) in two-dimension with respect to the x- and y- axes. The x-coordinate is called the abscissa, and the y-coordinate is called the ordinate.
What is meant by a quadrant?
In the Cartesian system, the coordinate plane is divided into four regions by the intersection of the x-axis and the y-axis. These regions are called quadrants.
What is the Cartesian coordinate system used for?
We use a three-dimensional Cartesian coordinate system to represent vectors. The Cartesian coordinate system is also used to represent points, lines, planes etc.
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