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This easy-to-understand and well-written article aims to help students understand the concepts of Analytical Geometry, Two dimensions (or 2D), and Two Dimensions – Rectangular Coordinate System in maths.
Coordinate geometry, also known as Analytical Geometry in mathematics, uses algebraic symbols and methods to represent and solve an equation for a problem in geometry.
Two Dimensions – Rectangular Coordinate System is expressed as the Cartesian coordinate system on a two-dimensional (i.e., 2D) plane that two perpendicular axes can define.
The concept of Analytical Geometry
Analytical Geometry is the study of geometry by using a coordinate system. It employs algebraic symbols and methods to represent and solve an equation for a geometrical problem.
Analytical geometry is commonly known as Coordinate geometry and sometimes as Cartesian geometry.
Analytical geometry establishes a correspondence between geometric curves and algebraic equations, making it possible to reformulate problems in geometry in terms of equivalent problems in algebra and vice versa.
Applications of Analytical Geometry:
The concept has uses in physics, engineering, aviation, rocketry, space science, and spaceflight.
It provides the foundation for most modern geometrical fields such as – algebraic geometry, differential geometry, discrete geometry, and computational geometry.
Two-Dimensional Coordinate System and Two Dimensions – Rectangular Coordinate System
A coordinate system provides a frame of reference for describing a system that one wants to analyse. The concept of Orthogonal coordinate systems is pretty popular in coordinate geometry. Here, orthogonal means ‘perpendicular’.
In two-dimensional coordinate geometry, unique coordinates represent every point in two-dimensional space. These coordinates identify the point in the plane or graph. We can find the coordinates of a point about two perpendicular reference lines known as coordinate axes.
The Two Dimensions – Rectangular Coordinate System involves two lines intersecting right angles. We can express it as the Cartesian coordinate system on a two-dimensional (i.e., 2D) plane that two perpendicular axes can define.
In Two Dimensions – Rectangular Coordinate System, the horizontal axis line is represented by X-axis, whereas Y-axis represents the vertical axis line. The values represented on the X and Y axes are real numbers.
Any point on the X-axis towards the left of the origin is negative. On the other hand, any point towards the right is a positive integer. Similarly, any point on the Y-axis below the origin is negative, while any above the origin is positive.
Some important terms to remember in the Two Dimensions – Rectangular Coordinate System are:
Axes of Coordinates:
In Two Dimensions – Rectangular Coordinate System, axes of coordinates are two intersecting straight lines used as reference lines in a graph or plane. For a plane with X and Y axes and origin at O, OX and OY together form the axes of coordinates representing the X-axis and Y-axis, respectively.
Origin and its Coordinate:
The point of intersection of the X and Y axes is known as the origin of that plane and is denoted by ‘O’. The coordinate of an origin in a Cartesian plane is (0, 0).
Abscissa and Ordinate:
The abscissa is the distance of any point on a plane from the Y-axis. On the other hand, the distance of any point from the X-axis is the ordinate.
Quadrant:
The X and Y axes divide the Cartesian plane into four equal parts known as quadrants. A quadrant is one-fourth of the plane divided by coordinate axes.
Some crucial formulas to remember in 2-D coordinate geometry are:
Distance between Two Points A and B:
Considering two points A (x1, y1) and B (x2, y2) on the Cartesian plane, the distance between them ‘d’, is given by the expression:
d = √ [(x2 – x1)2 + (y2 – y1)2]
Section Formula:
Considering two points A (x1, y1) and B (x2, y2) on the cartesian plane and P (x, y) is a point on line segment AB, such that AP: PB = m:n. Then, the coordinates of the point P are given by:
Case 1: P divides AB internally into m:n.
P (x, y) = [(mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)]
Case 2: P divides AB externally into m:n.
P (x, y) = [(mx2 – nx1) / (m – n), (my2 – ny1) / (m – n)]
Case 3: P divides AB in equal ratio such that m = n, i.e., P is a mid-point of line segment AB.
P (x, y) = [(x2 + x1) / 2, (y2 + y1) / 2]
Area of Triangle:
The area of a triangle ABC with vertices A (x1, y1), B (x2, y2), and C (x3, y3) is given by:
Area of /\ = ½ [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)].
Conclusion
This article on Analytical Geometry discusses the concept of two dimensions – Rectangular coordinate system. It helps us understand and grasp the concepts of analytical or coordinate geometry and its applications, two-dimensional (or 2-D) coordinate system and rectangular coordinate system, the related terms, and formulas involved in its concept.
We now know that we can express the Two Dimensions – Rectangular Coordinate System as the Cartesian coordinate system on a two-dimensional (2D) plane that two perpendicular axes can define. It involves two lines intersecting at a right angle.
