Formulas in Cartesian Coordinate System
We know that Cartesian Coordinate System is used to locate points and draw graphs for various algebraic function. Hence, the distance between the points and the equations for the graphs can be written using Cartesian System.
Distance Formula
Distance Formula is used to calculate distance between two points, two lines, between a point and a line and many more. The most commonly is used to calculate distance between two points in 2D and as well as three 3D. These formulas are mentioned below:
- Distance Formula for Two Points in 2D: √{(x2 – x1)2 + (y2 – y1)2}
- Distance Formula for Two Points in 3D: √{(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2}
Section Formula
Section formula is given to find the coordinates of a point which divides a given line in a given ratio.
Consider a line which is formed by joining two points (x1, y1) and (x2, y2) is divided by a Point P(x, y) in the ratio m:n then the coordinates will be given by
x = (mx2 + nx1)/(m + n) and y = (my2 + ny1)/(m + n)
Mid-Point Formula
In case of section formula if the ratio becomes equal i.e. 1:1 then it is called Midpoint Formula. Hence, if a Point is mid-point of a line then its coordinates are given as
x = (x1 + x2)/2 and y = (y1 + y2)/2
Slope of a Line
Slope of a line is the inclination of line with respect to the coordinate axes. The slope of a line is calculated as m = Tan θ where θ is the angle between line and the coordinate axis.
The formula for slope of line in cartesian form is given as
m = (y2 – y1)/(x2 – x1)
We know that Cartesian Coordinate System can also be used to draw graph for various algebraic expressions. In this article we will learn Cartesian Coordinate Equation of line and plane.
Equation of Line in Cartesian Form
The standard equation of a line is given by a linear equation epressed as ax + by + c = 0. However there are other forms also in which the equation of a line can be given. These equation are mentioned below:
- Slope-Intercept Form of Line: y = mx + C where m is the slope and C is the intercept.
- Intercept Form of Line: x/a + y/b = 1
- Point-Slope Form of Line: (y – y1) = m(x – x1)
- Two Point Form of Line: (y – y1) = {(y2 – y1)/(x2 – x1)}(x – x1)
- Normal Form of Line: L = x.cos θ + y.sin θ
Equation of Plane in Cartesian Form
A plane is a two dimensional flat region bounded by two coordinate axes. The different equations of Plane in cartesian form is given as follows:
Equation of Plane in Normal Form: [Tex]\vec r . \hat n = d [/Tex] where d is the perpendicular distance from the origin and n is the unit vector on the plane.
Equation of Plane Passing through three Non Collinear Points: [Tex](\vec r – a)[(\vec b – \vec a)\times(\vec c – \vec a)] = 0 [/Tex] where a, b and c are non-collinear points.
Eqution of Plane passing through intersection of Two Planes: If a plane pass through through intersection of two planes whose equation is given as [Tex]\vec r . \hat n_1 = d_1 [/Tex] and [Tex]\vec r . \hat n_2 = d_2 [/Tex] then its equation is given as [Tex]\vec r ( \hat n_1 + \lambda \hat n_2) = d_1 + \lambda d_2 [/Tex]
Cartesian Coordinate System
Cartesian Coordinate System in Maths is a division of coordinate geometry where the location of a point in a plane or space is marked by a pair of numbers or three numbers. The branch of geometry that deals with the Cartesian Coordinate System is called Coordinate Geometry. The numbers which are used to identify the location of a point in a plane or space are called coordinates.
The concept of the Cartesian Coordinate System is fundamental for class 9 and class 10 students which will later help them to understand various graphs and solve problems in physics.
In this article, we will study about the Cartesian Coordinate System, cartesian coordinates, coordinate axes, two dimensional as well as three-dimensional coordinate systems in detail.
Table of Content
- What is Cartesian Coordinate System in Maths?
- History of Cartesian Coordinate System
- Components of the Cartesian Coordinate System
- Cartesian Coordinates
- Coordinate Axes
- Cartesian Plane
- Dimension of Coordinate System
- One Dimensional Coordinate System
- Two Dimensional Coordinate System
- Three Dimensional Coordinate System
- How to Plot Points in Cartesian System of Coordinates?
- Formulas in Cartesian Coordinate System
- Distance Formula
- Section Formula
- Mid-Point Formula
- Slope of a Line
- Equation of Line in Cartesian Form
- Equation of Plane in Cartesian Form
- Cartesian Representation of Complex Numbers
- Application of Cartesian Coordinate System
- Cartesian Coordinates System Examples
- Cartesian Coordinate System Class 9
- Cartesian Coordinate System Questions