Basic Concepts Related to Cartesian System: The Cartesian plane is introduced to locate the position of an object in a two-dimensions. The Cartesian Plane, also known as the \(x-y\) plane or coordinate plane, is used to plot data pairs on a two-line graph. The Cartesian plane is named after Rene Descartes, a mathematician who first proposed the concept. Two perpendicular number lines intersect to form Cartesian planes.

Points on the cartesian plane are referred to as ordered pairs, which are extremely useful when illustrating the solution to equations with multiple data points. Simply put, the Cartesian plane is made up of two number lines, one vertical and one horizontal, that form right angles with one another. Continue reading this article regarding basic concepts related to cartesian system, its introduction, parts, quadrants, plotting points and more.

Need for Cartesian Coordinate System           

There are many real-life situations where we want to locate the position of an object. Let us say we need the position of a football on the ground or a house in the village on a map. In all these instances, we require more than one measure to locate the objects exactly. To ease such tasks, we take the help of cartesian planes.

Understanding a concept like the coordinate plane frequently requires putting abstract terminology and descriptions into context. Mathematics describes the real world, but it is not always clear how the concepts apply in practice. Coordinate planes can range from abstract representations of other variables to spatial coordinates with readily available real-world examples. To use a coordinate plane, simply select the type of system you want to use and specify the directions it should go in.

What is a Cartesian System?

A coordinate plane or a cartesian plane is a two-dimensional number line in which the vertical line is referred to as the \(y-\)axis, and the horizontal line is referred to as the \(x-\)axis. These lines are perpendicular to one another and cross at their zero points. This is referred to as the origin.

Quadrants on a Cartesian Plane

The coordinate axes divide the Cartesian plane into four quadrants.

• The first quadrant is at the top right section of the plane.
• The second quadrant is at the top left section of the plane.
• The third quadrant is at the bottom left section of the plane.
• The fourth quadrant is at the bottom right section of the plane.

A coordinate system is used to find the location of a point in a plane by using two perpendicular lines. Points are represented in two dimensions by coordinates \((x, y)\) with respect to the \(x-\)and \(y-\)axes, respectively.

Let us consider an example. Suppose Preethu’s house is in the \(5^{\text {th }}\) column and \(6^{\text {th }}\) row of the housing colony. Then, this position can be represented as \((5,6)\).

Parts of a Cartesian Plane

• Centre:

The intersection of the axes in the Cartesian plane is the origin, denoted by the letter \(O\), and its coordinates are denoted by \((0,0)\). To locate any point \(P\) in the plane, we measure the distance \(x\) along \(X-\) axis followed by the distance \(y\) parallel to \(Y-\) axis, to reach from \(O\) to \(P\).

• Axes:

• If we move to the right along the horizontal axis, \(x\) will be positive, and it will be negative, when we move to the left of the origin.
• Similarly, if moved down below the origin on the vertical axis, \(y\) value will be negative, and it will be positive above the origin.

• Coordinates:

The two real numbers \(x\) and \(y\) plotted together will uniquely describe the point \(P\).

The pair of real numbers will be different for different positions of \(P\).

• Abscissa: The \(x\)-coordinate of a point is the perpendicular distance from the \(y\)-axis. It is measured along the \(x\)-axis. This \(x\)-coordinate is called the abscissa.
• Ordinate: The \(y\)-coordinate of a point is the perpendicular distance from the \(x\)-axis. It is measured along the \(y\)-axis. This \(y\)-coordinate is called the ordinate.

Signs in the Quadrants

The sign convention for the coordinates of the point in:

• The first quadrant is \((+x,+y)\)
• The second quadrant are \((-x,+y)\)
• The third quadrant is \((-x,-y)\)
• The fourth quadrant is \((+x,-y)\)

These signs of Cartesian coordinates are shown in the image below.

How to Plot Points on a Cartesian Plane?

Let us plot a point whose Cartesian coordinate is \((2,3)\).

Step 1: First, identify the abscissa (\(x-\) value) and ordinate \((y-\) value\()\) from the given ordered pair.

To plot the point \((2,3)\) in the cartesian plane,

• Abscissa \(=2\)
• Ordinate \(=3\)

Step 2: On the \(x\)-axis, locate the value of \(x\) on the \(x\)-axis. Draw a line at the point perpendicular to the \(x\)-axis.

Here, \(x=2\). Move your pencil \(2\) steps to the right of the origin as \(x\) value is positive. Draw the line.

Step 3: On the \(y-\) axis, locate the value of \(y\). Draw a line at the point perpendicular to the \(y\)-axis.

Here, \(y=3\). Move the pencil \(3\) units up from the origin. We should move up as \(y\)-coordinate takes a positive value. Draw the line.

Step 4: The point where the two lines intersect is the required point.

Here, the coordinate is \((2,3)\), and the point lies in the first quadrant of the cartesian plane, as both \(x\) and \(y\) coordinates are positive.

Real-World Applications of the Cartesian Coordinate System

Following are few areas where the Cartesian Coordinate System is applicable,

• The position of any object in the real world can be described using the Cartesian coordinate system.
• The Cartesian coordinate system is used in military applications. The Cartesian coordinates can be used to determine the precise position on the plane.
• A projected coordinate system is intended to be used on a flat surface, such as a printed map or a computer screen.
• A coordinate framework is used to describe the correct location and shape of features for dealing with real-world locations. A geographic coordinate system is used to assign geographic locations to objects.

Solved Examples – Basic Concepts Related to Cartesian System

Below are a few solved examples that can help in getting a better idea.

Q.1. Locate the point \((2,3)\) on the cartesian coordinate system and identify the quadrant in which it is located.
Ans: To mark the point \((2,3)\):

• Take a horizontal distance of \(2\) units in the positive \(x-\)direction
• A vertical distance perpendicular from the horizontal reference axis as \(3\) units in the positive \(y-\)direction.

The point is marked as a blue dot in the figure.

We can see that \((2, 3)\) lies in the first quadrant as the \(x\) and \(y\) coordinates are positive.

Q.2. A square has one of its vertices at the origin, and the adjacent sides of the square are coincident with positive coordinate axes. If the side is of length \(5\) units, then what will be the other vertices?
Ans: As the sides of the square are coincident with the positive coordinate axes, the square will be lying in the first quadrant.
As the measure of each side is \(5\) units, the vertices will be \((5,0),(5,5)\) and \((0,5)\).

Q.3. Segregate the points according to the quadrants in which they lie: \(P(-7,6), Q(7,-2), R(-6,-7), S(3,5)\) and \(T(3,9)\)
Ans: The sign convention for the coordinates of the point in the:

• first quadrant are \((+x,+y)\)
• second quadrant is \((-x,+y)\)
• third quadrant is \((-x,-y)\)
• fourth quadrant is \((+x,-y)\)

Here, we can see that points \(S(3,5)\) and \(T(3,9)\) have both the coordinates positive. Hence, they lie in the first quadrant.
As, \(x-\)coordinate is negative and \(y-\)coordinate is positive, \(P(-7,6)\) lies in the second quadrant.
Similarly, point \(R(-6,-7)\) has both negative coordinates, and hence, lies in the third quadrant
The point \(Q(7,-2)\) lies in the fourth quadrant.

Q.4. Do any of the following points lie on the coordinate axes?
\(A(0,4), B(3,-1), C(1,0), D(1,2)\) and \(E(-3,0)\)
Ans: We know that if the points are on the coordinate axes, then any one of the coordinate values will be zero.
If the point is on \(x-\)axis, then \(y-\)value will be \(0\), and if the point lies on the \(y-\)axis, then the \(x-\)value will be \(0\).
Hence, \(A(0,4), C(1,0)\), and \(E(-3,0)\) lie on the coordinate axes.

Q.5. Do \(P(-3,2)\) and \(Q(3,-2)\) lie on the same quadrant?
Ans: No, the points \(P\) and \(Q\) do not lie in the same quadrant.
For point \(P(-3,2)\), the \(x-\)coordinate is negative, and the \(y-\)coordinate is positive. Hence, it lies in the second quadrant.
Whereas \(Q(3,-2)\) has a positive \(x-\)coordinate and a negative \(y-\)coordinate. Hence, \(Q\) lies in the fourth quadrant.

Summary

Two perpendicular lines, called axes, are used to locate the position of an object on a plane. The plane on which we locate the points is called the Cartesian plane or the coordinate plane. The lines are known as coordinate axes. The horizontal line is known as the \(x-\)axis, and the vertical line is known as the \(y-\)axis.

These two coordinate axes divide the plane into four sections called quadrants. The origin is the point at which the axes intersect. A point’s \(x-\)coordinate, or abscissa, is its distance from the \(y-\)axis, and its \(y-\)coordinate, or ordinate, is its distance from the\(x-\)axis. If the abscissa of a point is \(x\) and the ordinate is \(y\) then \((x, y)\) are called the coordinates of the point.

Frequently Asked Questions (FAQs)

Students might be having many questions with respect to the Basic Concepts Related to Cartesian System. Here are a few commonly asked questions and answers.

Q.1. What is the Cartesian system, and explain it?
Ans: In the Cartesian coordinate system, the location of a point is given by coordinates. These coordinates represent its distance from perpendicular lines intersecting at the origin.

Q.2. What is the importance of learning the concepts of the Cartesian coordinate system?
Ans: By understanding the cartesian coordinate system, we can easily locate the position of any object in the two-dimensional plane.

Q.3. Who has given the concept of the Cartesian system?
Ans: The Cartesian plane is named after René Descartes \((1596–1650)\), a French mathematician and philosopher who introduced the coordinate system to demonstrate how algebra could be used to solve geometric problems.

Q.4. How many quadrants are there in a Cartesian system?
Ans: There are four quadrants in a Cartesian system.

• Quadrant I
• Quadrant II
• Quadrant III
• Quadrant IV

Q.5. What is the sign convention in the Cartesian plane?
Ans: The sign convention for the coordinates of the point in the:

• First quadrant is \((+x,+y)\)
• Second quadrant is \((-x,+y)\)
• Third quadrant is \((-x,-y)\)
• Fourth quadrant is \((+x,-y)\)

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