Coordinate Axes are the fundamental concept in Coordinate Geometry, which is the combination of two branches of mathematics, i.e., algebra and geometry. The basic idea of coordinate geometry is to study the properties of various figures in mathematics, i.e., points, lines, curves, etc.
In any plane, we need a certain number of lines to represent the position of a point in the plane. These lines are called coordinate axes. For example, we need 2 coordinate axes to represent a point in the 2D plane. The plane that is formed by the intersection of these axes is called the Cartesian plane.
Coordinate Axes in Two Dimensions
In a 2-dimensional plane, a point is located using its distance from the two coordinate axes. These axes are perpendicular to each other always. Thus, coordinate axes in two dimensions help us determine the position of the point in terms of the horizontal and vertical distances of the point on each coordinate axis.
There are two axes in the two-dimensional coordinate system,
- X-axis
- Y-axis
X-axis and Y-axis
Cartesian coordinate system in 2 dimensions uses two axes to show a point in 2-dimensional space. These axes are X-axis and Y-axis . These axes are placed perpendicular to each other. The X-axis is the horizontal axis, and Y-axis is the vertical axis. The point of intersection of the X-axis and Y-axis is called Origin.
Each point in the cartesian coordinate system is determined using its x and y coordinate and represented as (x, y). The Origin is represented using O(0, 0). The X-axis is divided into two parts by the origin i.e., OX and OX'. OX' is used to show numbers with negative X coordinate. Similarly Y-axis is divided into two parts by the origin i.e. OY and OY'. OY' is used to show numbers with negative Y coordinate.
Quadrants and Sign Convention
The XX' and YY' intersect at origin and this results in the formation of 4 quadrants as shown in the below diagram:
1) The region XOY is the First Quadrant.
- In first quadrant, both x and y coordinate are positive i.e. (+, +).
2) The region X'OY is the Second Quadrant.
- In second quadrant, x coordinate is negative and y coordinate is positive i.e. (-, +).
3) The region X'OY' is the Third Quadrant.
- In third quadrant, both x and y coordinate are negative i.e. (-, -).
4) The region XOY' is the Fourth Quadrant.
- In fourth quadrant, x coordinate is positive and y coordinate is negative i.e. (+, -).
Representing a Point On Coordinate Axes
A point is represented on coordinate axes using its x coordinate and y coordinate. The X coordinate is called abscissa and the Y coordinate is called ordinate. A point P is represented as P(x, y) as shown in the below figure:
In the above figure point P is shown on the Coordinate axes system. The length OR represents the abscissa and length OS represents ordinate of point P.
If the abscissa of a point is zero then it will lie on the y-axis (YY') and if the ordinate of a point is zero then it will lie on x-axis (XX').
Coordinate Axes in Polar Coordinate System
In polar coordinate system , the point is represented using two values which are its distance from the origin or the reference point and an angle from the reference direction. Thus the system that shows a point by its distance and angle from a reference is called polar coordinate system.
Polar Axes and Origin
In the polar coordinate system, origin is termed as pole. Compared to the normal coordinate grid, the polar coordinate system uses a series of concentric circles around the pole. The axes are represented similarly as in the normal coordinate system. A point in polar coordinate system is represented as P(r, θ) where r is the distance of point from the pole and θ is the angle made by the line joining the point and the pole with the reference direction. The concentric circles have a difference of 1 unit in their radius.
Coordinate Axes in Three Dimensions
We have learnt about representing a point in 2D plane but in real life we generally deal with 3D planes. Hence we need an additional parameter i.e. height of the point to locate it in the 3D plane. Thus we make use of 3 coordinate axes to represent the point in the plane.
X-axis, Y-axis, and Z-axis
Cartesian coordinate system in 3 dimensions uses three axes to show a point in 3 dimensional space. These axes are X-axis, Y-axis and Z-axis. Similar to 2D plane, these axes are placed mutually perpendicular to each other. X-axis is the horizontal axis and Y-axis is the vertical axis. Z-axis is generally represented as the third axis mutually perpendicular to the X and Y axis. The point of intersection of X-axis, Y-axis and Z-axis in 3D plane is called the Origin.
Each point in the 3D cartesian coordinate system is determined using its x, y and z coordinate and represented as (x, y, z). The Origin is represented using O(0, 0, 0). The X-axis is divided into two parts by the origin i.e. OX and OX'. OX' is used to show numbers with negative X coordinate. Similarly Y-axis is divided into two parts by the origin i.e. OY and OY'. OY' is used to show numbers with negative Y coordinate. Similarly Z-axis is divided into two parts by the origin i.e. OZ and OZ'. The origin also divides the entire 3D plane in eight equal parts called octants.
Coordinate Axes in Other Coordinate Systems
Apart from all these coordinate axes systems, there are other coordinates system also. These are:
- Cylindrical Coordinate System
- Spherical Coordinate System
Cylindrical Coordinate System
Cylindrical coordinate system is a combination of the normal coordinate system and the polar coordinate system. a point in cylindrical coordinate system is represented using three quantities i.e. radial distance r, azimuthal angle θ, and the height along the z-axis. Thus a point P is represented as P(r, θ, z).
The radial distance r is the distance of the projection of point P (say the projection is represented using Q) in the XY plane from the origin. Azimuthal angle θ is the angle between X-axis and the line joining origin and Q. Height z is the distance of the point from XY plane.
Spherical Coordinate System
Cylindrical coordinate system is used to represent a point on 3D plane. A point in spherical coordinate system is represented using three quantities i.e. radial distance r, polar angle θ, and the azimuthal angle φ. Thus a point P is represented as P(r, θ, φ).
The radial distance r is the distance of the point P from the origin. In spherical coordinates, θ (polar angle) = angle with positive Z-axis. Azimuthal angle φ is the angle with positive X-axis in XY plane.
Applications of Coordinate Axes System
The system of coordinate axes finds application in various fields such as:
- It is used to locate a place on the surface of the Earth in the form of longitudes and latitudes.
- It is also used in navigation system such as Google Maps to determine the coordinates of a person and the places.
- It can be used to location of one object with respect to the other.
Solved Problems on Coordinate Axes
Problem 1: Identify the quadrant in which the point A(5, -4) lie.
As the abscissa of the point is positive and ordinate is negative, the point A(5, -4) will lie in fourth quadrant.
Problem 2: How will a point having abscissa as 2 and ordinate as 7 and lying in third quadrant be represented?
As the point lies in third quadrant, abscissa and ordinate both will be negative. Thus the point will be represented as (-2, -7).
Problem 3: On which axis will a point having abscissa as 4 and ordinate as 0 and lying in first quadrant lie?
As the point has ordinate as 0, it means that the point will lie on XX' i.e. X-axis.
Problem 4: Identify the quadrant in which the point A(1, 1) lie.
As the abscissa of the point is positive and ordinate is positive, the point A(1, 1) will lie in first quadrant.
Problem 5: How will a point having abscissa as 8 and ordinate as 3 and lying in second quadrant be represented?
As the point lies in second quadrant, abscissa will be negative and ordinate will be positive. Thus the point will be represented as (-8, 3).
Practice Questions on Coordinate Axis
Problem 1: Identify the quadrant in which the point B(−3, 5) lies.
Problem 2: How will a point having abscissa as -2 and ordinate as -6, Can this point lie in the second quadrant? If not, in which quadrant does it lie?
Problem 4: Identify the quadrant in which the point C(−4, −3) lies.
Problem 5: How will a point having abscissa as -8 and ordinate as 3. Can this point lie in the fourth quadrant? If not, in which quadrant does it lie?