Cartesian System

Cartesian Coordinate System

Co-ordinate system is used to locate the position of a point in a plane using two perpendicular lines. Points are represented in the form of coordinates (x, y) in two-dimension with respect to x- and y- axes. In this article, we will learn about Cartesian Coordinate system.To understand the need of coordinate system, let us consider an example, suppose Rina is a girl in your class and she sits on the 3^rd column and 5^th row. Then, this position can be represented as (3, 5).

Two axes – vertical axis and perpendicular axis are reference lines of a rectangular system from which distances are measured. They are obtained as follows:

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.

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 generalised 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.

Example Problems

Example 1: Locate below points on the cartesian coordinate system. Also, mention the quadrant points belong to.

(i) (2, 3)

(ii) (-3, 1)

(iii) (-1.5, -2.5)

(iv) (0,0)

Solution:

Explanation:

(i) Point (2,3) marked by green dot in the graph. The horizontal distance is 2 units in positive x-direction and the vertical distance perpendicular from the reference horizontal axis is 3 units in positive y-direction and hence we denote the point as (2,3) in coordinate form and it belongs to quadrant I since both are positive. Similarly, we can extend this same idea to other points marked in the graph as well.

(ii) For the point marked in red colour, the horizontal distance is 3 units in negative x-direction and the vertical distance perpendicular from the reference horizontal axis is 1 unit in the positive y-axis and hence we denote the point as (-3,1) in coordinate form. It belongs to quadrant II as x-axis is negative and the y-axis is positive.

(iii) For the point marked in blue, the distance in horizontal and vertical units are respectively 1.5 and 2.5 both in negative directions. Thus, we have the point (-1.5.-2.5) in quadrant III.

(iv) Origin is the point of intersection of horizontal and vertical axes, which is (0, 0). It does not belong to any quadrant.

Example 2: If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle PQR is (are) always rational point(s)
A) Centroid

B) Incentre

C) Circumcentre

D) Orthocentre (A rational point is a point both of whose coordinates are rational numbers)

Solution: 

If 

where 

etc., are rational numbers then

are also rational. So, the coordinates of the centroid 

will be rational.

As 

may or may not be rational and it may be an irrational number of the form 

Hence, the coordinates of the incentre 

may or may not be rational.

If 

be the circumcentre or orthocentre, a and b are found by solving two linear equations in 

with rational coefficients. So 

must be rational numbers.

If the vertices of PQR are rational points, then the centroid, the orthocentre and circumcentre will always be rational points.
Therefore, Options A), B) and C) are correct,

Example 3: Let A(2, -3) and B(-2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line _____.

Solution: Let the third vertex be (x₁,y₁) then centroid 

 i.e., 

Given, centroid of triangle moves on the line

2x+3y =1

So 2(x₁/3)+3(y₁-2)/3 = 1

i.e.,  

Locus of (x₁,y₁) is 2x+3y=9

Example 4: One of the vertices of a square is origin and adjacent sides of the square are coincident with positive axes. If side is 5 then which will not be its one of the vertex

(A) (0, 5)

(B) (5, 0)

(C) (- 5, -5)

(D) (0, 0)

Solution:

The square lies in first quadrant. So (-5, -5) will not be a vertex.

Hence option C is the answer. 

Plotting a Point in the Plane if its Coordinates are given

In Mathematics, a cartesian plane is a two-dimensional plane, which is defined by two perpendicular lines, such as horizontal line and vertical line. The cartesian plane extends infinitely in both directions. Here, we are going to learn how to plot a point in the plane if its coordinates are given with many solved examples.

Introduction to Cartesian Plane

A cartesian plane is a two-dimensional coordinate plane defined by two perpendicular lines such as x-axis and y-axis. The horizontal number line is called x-axis and the vertical number line is called y-axis. The point of intersection of these two lines is called origin. The region of x-axis towards the right side of the origin is called positive x-axis and the region of x-axis towards the left side of the origin is called negative x-axis. Similarly, the region of y-axis above the origin is called positive y-axis and the region of y-axis below the origin is called negative y-axis. The point on the coordinate plane is represented by the ordered pair (x, y).

Rules to Plot a Point in the Coordinate Plane

The two important rules to plot a point in the cartesian plane are given below:

1. The first coordinate in the ordered pair (x) represents the left/right movement of a point from the origin.
2. The second coordinate in the ordered pair (y) represents the up/down movement of the point from the origin.

Solved Examples

Example 1:

Plot the coordinates (3, 5) and (5, -4) in the cartesian coordinate system.

Solution:

Step 1: Draw the coordinate axes, such as the x-axis and y-axis

Step 2: Choose the unit such that 1 centimetre represents one unit on both the x-axis and y-axis.

Step 3: The coordinate (3, 5) represents the distance from the origin to the positive x-axis is 3 units and the distance from the origin to the positive y-axis is 5 units. Now, mark these points on the coordinate plane, and name it as “P”.

Step 4: The coordinate (5, -4) represents the distance from the origin to the positive x-axis is 5 units and the distance from the origin to the negative y-axis is 4 units. Mark the coordinate point in the plane and label it as “Q”.

Step 5: Therefore, the point P (3, 5) lies in the first quadrant and the point Q (5, -4) lies in the fourth quadrant.

Example 2:

Plot the following points in the cartesian plane (Use the scale: x-axis = 1 cm and y-axis = 1 cm)

x	-3	0	-1	4	2
y	7	-3.5	-3	4	-3

Solution:

From the given table, the ordered pairs formed are (-3, 7), (0, -3.5), (-1, -3), (4, 4) and (2, -3).

The point (-3, 7) lies in the 2^nd quadrant

The point (0, -3.5) lies in the negative y-axis

The point (-1, -3) lies in the 3^rd quadrant

The point (4, 4) lies in the 1^st quadrant

The point (2, -3) lies in the 4^th quadrant.

Now, plot these points on the cartesian plane.

.Practice problems

1. Mark the points (5, 0), (0, 5), (2, 5), (5, 2), (–3, 5), (–3, –5), (5, –3) and (6, 1) in the cartesian coordinate plane.
2. Plot the points in the coordinate plane (– 2, 4), (3, – 1), (– 1, 0), (1, 2) and (– 3, – 5) and identify the quadrant or axis where each of the following points lies.

Frequently Asked Questions on Plotting a Point in the Plane if its Coordinates are given

What is a cartesian plane?

A cartesian plane is a coordinate plane that is defined by two axes, such as the x-axis and y-axis. The horizontal line is called the x-axis and the vertical line is called the y-axis.

Define origin.

The point of intersection of the x-axis and y-axis is called an origin.

What are abscissa and ordinate?

The distance of a point from the y-axis is called abscissa and the distance of a point from the x-axis is called ordinate.

What is the coordinate of origin?

The coordinate of origin is (0, 0).

In which quadrant is the point (2, 2) located?

A Point (2, 2) is located in the first quadrant.