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Cartesian Plane is a two-dimensional plane that is part of the cartesian coordinate system. The cartesian plane was invented by Rene Descartes in the 17th Century. The most important feature of a cartesian plane is that it links two fields of mathematics - namely, Euclidean Geometry and Algebra.
Any point on a cartesian plane is specified by numerical coordinates. The coordinates of a point on a cartesian plane are expressed as an ordered pair. Furthermore, these points are signed and are located at a fixed distance from two perpendicular lines known as axes. There can be two coordinate axes in a cartesian plane, namely, the x-axis and the y-axis. In this article, we will take an in-depth look at the definition, quadrants, and graphs of a cartesian plane.
|1.||What is Cartesian Plane?|
|2.||Cartesian Plane Quadrants|
|3.||Plotting Points on the Cartesian Plane|
|4.||Cartesian Plane Graph|
|5.||FAQs on Cartesian Plane|
What is Cartesian Plane?
A cartesian plane is part of the cartesian coordinate system. This coordinate system can be translated into one, two, and three dimensions. In two dimensions, the plane is called the cartesian plane. It can also be called the coordinate plane.
Cartesian Plane Definition
A cartesian plane can be defined as a plane formed by the intersection of two coordinate axes that are perpendicular to each other. The horizontal axis is called the x-axis and the vertical one is the y-axis. These axes intersect with each other at the origin whose location is given as (0, 0). Any point on the cartesian plane is represented in the form of (x, y). Here, x is the distance of the point from the y-axis and y is the distance from the x-axis.
Cartesian Plane Example
The two horizontal and vertical intersecting lines are the x and y axes respectively. The coordinates of the point (5, 6) indicate that it is located at a distance of 5 units from the y-axis and 6 units from the x-axis.
Parts of a Cartesian Plane
A cartesian plane can be divided into three major parts. These three parts are vital when we try to locate a point on the cartesian plane or draw the graph of a certain function. These are given below as follows:
Axes - The two lines that intersect to form the cartesian plane are known as the axes. The horizontal line is called the x-axis. The vertical line that is perpendicular to the x-axis is known as the y- axis.
Origin - The point where the two perpendicular axes - x and y meet is known as the origin. The coordinates of the origin are given by (0, 0). The axes are divided into two equal parts by the origin.
Quadrants - When the x and the y axes intersect, it divides the cartesian plane into 4 regions. These are known as quadrants and extend infinitely.
Cartesian Plane Quadrants
When the two axes intersect each other, it divides the cartesian plane into four infinite regions. These 4 regions are known as quadrants. The quadrants are bound by the two semi x and y axes. The quadrants can be numbered from one to four in an anti-clockwise direction. The signs of the x and the y coordinates of a point will be different in each coordinate. Depending on the value of a point, it can be located in a particular quadrant as given below.
- First Quadrant - x > 0 and y > 0. Thus, the sign of a point will be ( +, +).
- Second Quadrant - x < 0 and y > 0. Thus, the sign of a point will be ( -, +).
- Third Quadrant - x < 0 and y < 0. Thus, the sign of a point will be ( -, -).
- Fourth Quadrant - x > 0 and y < 0. Thus, the sign of a point will be ( +, -).
The positive direction will be upwards and towards the right while the negative direction is downwards and to the left.
Plotting Points on Cartesian Plane
All distances are measured from the origin (0, 0) when plotting points on a cartesian plane. The points are represented in the form of a signed ordered pair. The following steps are used to plot a point, P(7, -6) on the cartesian plane.
- Check the signs of the x and y coordinates of the point. Depending on the signs, identify the quadrant where the point will lie. As the signs of the point P(7, -6) are of the form (+, -), it will lie in the fourth quadrant.
- Using the x coordinate value move x spaces to the left (negative) or right (positive) on the x-axis from the origin. Draw a vertical perpendicular line from this point. The x coordinate of the point is 7 hence, move 7 spaces to the right from the origin on the x-axis and sketch a vertical line.
- Using the y coordinate value move y spaces towards the bottom (negative) or top (positive) on the y-axis from the origin. Draw a horizontal perpendicular line from this point. The y coordinate value is -6 thus, we move 6 spaces downwards from the origin. Now we sketch a horizontal line.
- The point of intersection of these two lines is our point P(5, -6).
If we have a point in the form of (0, y). It implies that the point lies on the y-axis at a distance of y from the x-axis.
Suppose our point is given by (x, 0). Then the point lies on the x-axis at a distance of x from the y axis.
How to Plot Complex Numbers in Cartesian Plane?
In addition to plotting real numbers, the cartesian plane can also be used to plot complex numbers. In such a case, the cartesian plane is known as the complex plane. The x-axis is used to depict the real part of the number and the y-axis denotes the imaginary part. The steps to plot a complex number are similar to that of plotting real numbers as given below:
- Let the complex number be of the form a + ib.
- We move 'a' spaces on the x-axis to the left or right depending upon the sign of a. Draw a vertical line from this point.
- Next, we move 'b' spaces upwards or downwards on the y-axis. Draw a horizontal line at this point.
- The point of intersection of these two lines will give us the complex number.
Cartesian Plane Graph
A cartesian plane graph is a diagram that gives the visual representation of the relationship between two variables. When we have an equation in two variables we require two axes (x and y) to graph it. The steps to graph an equation in two variables are as follows:
- Substitute the value of x with some numerical value.
- Find the corresponding value of y.
- Perform the first two steps multiple times to get a number of test points.
- Connect these points to get the required graph.
Using some standard equations we can trace out certain well-known shapes. The equations along with the shape of the graph are given below:
- y = mx + c. This is a linear equation in two variables and will trace out a straight line on the cartesian plane.
- (x - h)2 + (y - k)2 = r2. This equation will result in a circle with the center at (h, k) and the radius measures r units.
- y2 = 4ax. This is the standard equation of a parabola.
Thus, depending on the type of equation in two variables and the degree, different types of curves can be drawn on the cartesian plane.
Important Notes on Cartesian Plane
- A cartesian plane also called a coordinate plane is formed by the intersection of 2 perpendicular axes.
- There are four quadrants in a cartesian plane. The signs of the coordinates in each quadrant is given as (+, +) (first quadrant), (-, +) (second quadrant), (-, -) (third quadrant) and (+, -) (fourth quadrant).
- A graph of an equation in two variables can be traced out on the cartesian plane.
Examples on Cartesian Plane
Example 1: Plot the point (-3, 2) on a cartesian plane.
Solution: As the coordinate is of the form (-, +) hence, the point lies in the second quadrant. We move 3 spaces to the left of the origin and then 2 spaces upwards to get our point.
Example 2: State which quadrants the following points lie in.
a) (5.5, -1)
b) (-7, -3)
c) (2, 3.45)
Solution: a) As the (5.5, -1) is of the form (+, -), thus, it lies in the fourth quadrant.
b) (-7, -3) lies in the third quadrant as x < 0 and y < 0.
c) As the sign of both coordinates is positive hence, (2, 3.45) lies in the first quadrant.
Example 3: Find the coordinates of the following points on the cartesian plane:
Solution: Coordinates of A are (-3, 4). It lies in the second quadrant. Coordinates of B are (4, -3) and it lies in the fourth quadrant.
Practice Questions on Cartesian Plane
FAQs on Cartesian Plane
What is the Meaning of Cartesian Plane?
When two coordinate axes (x and y) intersect it forms a cartesian plane. These axes are always perpendicular to each other. The point of intersection of these two lines is known as the origin.
What are the Quadrants on a Cartesian Plane?
There are four quadrants in a cartesian plane. These quadrants are bound by the two semi-axes. The signs of points in various quadrants are given as (+, +) (first quadrant), (-, +) (second quadrant), (-, -) (third quadrant) and (+, -) (fourth quadrant).
How to Plot Points on a Cartesian Plane?
To plots points on a cartesian plane the steps are as follows:
- Identify the quadrant depending upon the sign of the ordered pair.
- Move x points to the left or right depending on the sign of the x coordinate.
- Now move y points above or below based on the sign of the y coordinate. This gives us the point.
Is Cartesian Plane a Number Line?
A cartesian plane is not a number line. A cartesian plane is the coordinate system plane in two dimensions. However, the coordinate system plane in one dimension will be a number line.
How to Find Coordinates on a Cartesian Plane?
To find the coordinates on a cartesian plane we first determine the distance of the point from the y-axis. This becomes the x coordinate. We then find its distance from the x-axis to give the y coordinate. The point's coordinates are represented as (x, y).
What is Cartesian Plane Used For?
The cartesian plane is used to draw graphs of equations in two variables. We can plot the data points according to the given equation to get its corresponding graph. This feature is heavily used in the analysis of data in various industries.
Do Cartesian Planes Extend Infinitely?
Yes, cartesian planes extend infinitely. They are formed by the intersection of two perpendicular lines. Thus, we get four quadrants that are bound by the semi-axes.