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.
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.
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.
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:
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.
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).
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.
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.
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]
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]
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)].
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.
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