Explanation:

Take two number lines XX’ and YY’. Place XX’ in horizontal and write the numbers on it as we write in the number line. Similarly, place YY’ in vertical and proceed writing numbers on it as we write in a number line. Combine both the lines in such a way that the two lines cross each other at their zeroes or origins. The horizontal line XX’ is called the x-axis and the vertical line YY’ is called the y-axis. The point where XX’ and YY’ cross is called the origin, and is denoted by O. Since the positive numbers lie on the directions OX and OY, OX and OY are called the positive directions of the x-axis and the y-axis respectively. Similarly, OX’ and OY’ are called the negative directions of the x- and y-axes respectively.

Important Terms:

Quadrants:

Moreover, the axes divide the plane into four parts and these four parts are called quadrants (one-fourth part). Thus, we have four quadrants numbered I, II, III and IV anticlockwise from OX.

Points in different Quadrants.

Signs of coordinates of points in different quadrants:

I Quadrant: ‘+’ x – coordinate and ‘+’ y – coordinate. E.g. (2, 3)

II Quadrant: ‘-’ x – coordinate and ‘+’ y – coordinate. E.g. (-1, 4)

III Quadrant: ‘-’ x – coordinate and ‘-’ y – coordinate. E.g. (-3, -5)

IV Quadrant: ‘+’ x – coordinate and ‘-’ y – coordinate. E.g. (6, -1)

Cartesian Plane:

A plane consists of axes and quadrants. Thus, we call the plane the Cartesian Plane, or the Coordinate Plane, or the x-y plane. The axes are called the coordinate axes.

Cartesian coordinate system for one dimensional:

The Cartesian coordinate system for one dimensional space consists of a line. We choose a point O, origin on the line, a unit of length and orientation for the line. The orientation chooses which of the two half lines determined by O is the positive, and which is negative. Each point P of the line can be specified by its distance from O, taken with a negative or positive sign.

Number line:

A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line. Every point on the number line can be interpreted as a number.

Important Note:

The above depicts a two-dimensional system. In case of a three-dimensional system, we have three mutually perpendicular axes, namely x, y and z. It can be generalized to create n coordinates for any point in n-dimensional Euclidean space.

Abscissa and Ordinate

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.

In writing the coordinates of a point in the coordinate plane, the x-coordinate comes first and then the y-coordinate. We place the coordinates in brackets as (x, y). The coordinates describe a point in the plane uniquely. It implies (3,1) ≠ (1,3) or in general (x, y) ≠ (y, x).

Consider an example point (5,6). Here abscissa = 5 and ordinate = 6.

Different Types of Coordinate Systems

We have 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 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.

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 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)