Cartesian System: Coordinate system, Equation, Graph, Examples

Cartesian System: We know how to describe the position of a point on the number line. There are many other situations when we need to find a point; we must define its position regarding more than one line. For example, put a dot on a sheet of paper. Can you tell the dot’s position on the paper? You may attempt to say the dot is in the top half of the paper” or “It is near the left boundary of the paper” Do these declarations fix the position precisely? No! But if you say the distance of a point with reference to edges, it gives some idea about the dot’s position.

This simple concept has provided a significant branch of mathematics known as Coordinate Geometry. This study was initially developed by mathematician Rene Descartes. In honour of Descartes, the system used for describing a point in the plane is known as the Cartesian system.

Cartesian Coordinates Definition

Cartesian coordinates allow one to determine the location of a point in the plane or three-dimensional space. The Cartesian coordinates (also called rectangular coordinates) of a point are a pair of numbers (in two-dimensions) or a triplet of numbers (in three-dimensions) that specify signed distances from the coordinate axis.

Cartesian Coordinate System

We use a number line to represent numbers by indicating points on the line at equal distances. Look at the following integer line.

It is observed that distances marked from both sides from a fixed point are called origin on the number line and denoted by \(0.\) All the positive numbers are shown on the right side of zero and all negative numbers on the left side. We are taking two number lines perpendicular to each other in the plane to locate a point regarding these two lines. Observe the following figure.

We sketch a horizontal number line and a vertical number line meeting at a point perpendicular to each other. The point of intersection is denoted as the origin. The horizontal number line \(XX’\) is known as \(X – axis\) and the vertical number line \(YY’\) is known as \(Y – axis\).

The point where \(XX’\) and \(YY’\) intersect is called the origin and is denoted by \(O\). Subsequently, the positive numbers lie on the directions \(OX,\) the positive direction of the \(X – axis\) similarly \(OY’\) is the positive \(Y – axis\) respectively. Also \(OX’\) and \(OY’\) are called the negative directions of the \(X – axis\) and the \(Y – axis\) We observe that the axes divide the plane into four parts.

These four parts are called the cartesian coordinate system quadrants and are denoted by \({Q_1},\,\,{Q_2},\,\,{Q_3}\) and \({Q_4}\) in an anti-clockwise direction. The plane here is called the cartesian plane or co-ordinate plane or \(XY – \) plane. The axes are called the coordinate axes.

Cartesian System of Rectangular Coordinates

The rectangular coordinate system consists of two integer lines that intersect at a right angle. The horizontal number line is called the \(x – axis\) and the vertical number line is called the \(y – axis\). These two-number lines describe a smooth surface called a plane, and every point on this plane has been associated with an ordered pair of real numbers \((x,y)\).

The first number is called the \(x – \) coordinate or abscissa, and the second number is called the \(y – \) coordinate or ordinate. The junction of the two axes is known as the origin, which corresponds to the point \((0,0)\).

An ordered pair \((x,y)\) represents the position of a point relative to the origin. The \(x – \) coordinate is representing a position towards the right of the origin if it is positive and to the left of the origin if it is negative. The \(y – \) coordinate represents a position above the origin if it is positive and below the origin if it is negative.

If you use this system, every position (point) in the plane is uniquely identified. For example, the pair \((5,\,2)\) denotes the position relative to the origin as shown:

Cartesian Equation

Any curve can be represented by having an equation in \(x\) and \(y\) describing the relation between \(x\) and \(y\) coordinates at each point of the curve. Such equations are called the Cartesian form of an equation.

For example, \({x^2} + {y^2} = 4\)

Now, when \(y = 0,\,\,{x^2} + {0^2} = 4 \Rightarrow {x^2} = 4 \Rightarrow x =  \pm 2\) and when \(x\, = \,0\,,\,{0^2} + {y^2} = 4 \Rightarrow {y^2} = 4 \Rightarrow y =  \pm 2\)
So, coordinates are \((2,0),\,( – 2,\,0),\,(0,\,\,2)\,and\,(0,\, – 2)\)

By plotting the points \((2,0),\,( – 2,\,0),\,(0,\,\,2)\,and\,(0,\, – 2)\) on cartesian plane, we get

We can see that the Cartesian form of the equation \({x^2} + {y^2} = 4\) represents a circle.

Cartesian to Polar Coordinates

In two dimensions, the Cartesian coordinates \((x,y)\) specify the location of a point \(P\) in the plane. The other two-dimensions in the coordinate system are polar coordinates. Rather than using the determined distances along the two coordinate axes, polar coordinates determine the location of a point \(P\) in the plane by its distance \(r\) from the origin and the angle \(\theta \) made between the line segment from the origin to \(P\) and the positive \(x – {\rm{axis}}{\rm{.}}\) The polar coordinates \((r,\,\,\theta )\) of a point are illustrated in the below figure.

As \(r\) ranges from \(0\) to infinity and \(\theta \) ranges from \(0\) to \(2\pi \), the point \(P\) specified by the polar coordinates \((r,\,\theta )\) covers every point in the plane. Adding \(2\pi \) to \(\theta \) brings us back to the same point, so if we allow \(\theta \) to range over an interval larger than \(2\pi \) each point would have multiple polar coordinates.

Hence, we typically restrict \(\theta \) to be in the interval \(0\,\, \le \,\theta \, < \,2\pi .\) Even with that limitation, there still is some non-exceptionality of polar coordinates when \(r = 0\) and the point \(P\) is at the origin independent of the value of \(\theta \).

\(r = \,\sqrt {{x^2} + {y^2}}\) and \(\theta \, = \,\arctan \frac{y}{x}\)

Polar Coordinates to Cartesian

We can compute the Cartesian coordinates of a point with polar coordinates \((r,\,\theta )\) by forming the right triangle shown in the below figure. The hypotenuse is the line segment from the origin to the point, and its length is \(r\).

The forecast of this line segment on the \(x – axis\) the leg of the triangle adjacent to the angle \(\theta \), so \(x = r\cos \theta .\) The \(y – \) component is determined by the other side, so \(y = r\sin \theta \).

Cartesian Plane Graph

Take into account the following diagram in which the axes are drawn on graph paper. Let us look at the distances of the points \(B\) and \(M\) from the axes. For this, we draw perpendiculars \(BC\) on the \(x – axis\) and \(BA\) on the \(y – axis.\) Similarly, we draw perpendiculars \(MN\) and \(MP\) as shown in the figure.

1. The perpendicular distance of the point \(B\) from the \(y – axis\) measured along the positive direction of the \(x – axis\) \(AB\, = \,OC\, = 4\) \({\rm{units}}\).
2. The perpendicular distance of the point \(B\) from the \(x – axis\) measured along the positive direction of the \(y – axis\) is \(BC\, = \,OA\, = 3\,{\rm{units}}\)
3. The perpendicular distance of the point \(M\) from the \(y – axis\) measured along the negative direction of the \(x – axis\) is \(MP\, = \,NO\, = 6\,{\rm{units}}\).
4. The perpendicular distance of the point \(M\) from the \(x – axis\) measured along the negative direction of the \(y – axis\) is \(MN = \,PO\, = 2\,{\rm{units}}\).

Note: The \(x – \) coordinate is also called the abscissa, and the \(y – \)coordinate is also called the ordinate.

Cartesian Equation of a Plane

In geometry, the word “normal” defines an object, such as a vector or line that should be perpendicular to the given object. Here, we can learn the equation of a plane in the normal form. It can be determined if two factors are known. One is normal to the plane, while the other is the plane’s distance from the origin.

Equation of the plane in normal and cartesian form

Vector form of an equation of the plane in normal form is \(\overrightarrow r .\widehat n\,\, = \,d\)

Where \(\overrightarrow r \) is the location where the vector of the point in the plane, \(\widehat n\) is the normal unit vector alongside the normal merging the origin to the plane, and \(d\) is the perpendicular distance of the plane from the origin.
Let \(P(x,\,y,\,z)\) be every point on the plane, and \(O\) is the origin. Then, we have,

\(\overrightarrow {OP}  = \,\overrightarrow r  = x\widehat i + y\widehat j + z\hat k\)

Now, the direction cosines of \(\widehat n\) as \(l,\,m\) and \(n\) given by \(\widehat n\, = \,l\widehat i + m\widehat j + n\hat k\)
Fromthe equation \(\overrightarrow r .\widehat n\, = d\), we get \((x\widehat i + y\widehat j + z\hat k).(l\widehat i + m\widehat j + n\hat k) = d\)

Therefore, the Cartesian form of an equation of the plane in normal form is \(lx\, + \,my\, + nz\, = d\).

Uses of Cartesian Coordinate System in Real Life

1. The position of any object in the real world can be described using a simple coordinate system.
2. Cartesian Coordinate system finds use in military service. A target’s precise position can be determined by using the cartesian coordinates.
3. A projected coordinate system is designed for a flat surface, such as a printed map or a computer screen.
4. 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 – Cartesian System

Q.1. Write the following
a) Coordinates of the point \(B\)
b) The abscissa of the point \(D\)

Answer: From the given cartesian system, we get
a) The distance of the point \(B\) from the \(X – {\rm{axis}}\) is \(4\,{\rm{units}}\), and the distance of the point \(B\) from the \(Y – {\rm{axis}}\) is \(1\,{\rm{unit}}\).
Therefore, the coordinates of the point \(B\) are \((4,\,1)\).
b) \(x – \) coordinate of ordered pair is also known as abscissa.
Here, the point \(D\) is \((2,\, – 2)\) and the abscissa of the point \(B\) is \(2\).

Q.2. Locate the points \((2,\,0),(0,3),( – 4,5),( – 4,\, – 5),(2, – 3),(3, – 5)\) and \((5,\,1)\) in the Cartesian plane.
Ans: Taking \(1\,{\rm{cm}}\,\,{\rm{ = }}\,\,{\rm{1}}\,{\rm{unit}}\) we draw the \(x\, – \,axis\) and the \(y\, – \,axis\). Dots show the position of the points in the graph given below:

Q.3. Plot the points \(( – 3,\,3),\,(2,3),( – 1, – 6)\) and \((5, – 1)\) in the coordinate plane and Join them in order to determine the geometrical shape formed.
Ans: Taking \(1\,{\rm{cm}}\,{\rm{ = }}\,1\,{\rm{unit}}\), we draw the \(x – axis\) and the \(y – axis\) Dots show the position of the points in the graph given below:

The geometrical shape obtained by joining points is a quadrilateral.

Q.4. Identify the quadrants in which the following points lie.\(P( – 7,6),Q(7,\, – 2),R( – 6, – 7),S(3,5)\) and \(T(3,9)\).
Ans: We know that,
1. In the first quadrant, both \(x\) and \(y\) coordinates are positive so, here points \(S(3,\,5)\) and \(T(3,\,9)\)  lies in the first quadrant.
2. In the second quadrant, \(x\) coordinate is negative, and \(y\) coordinate is positive so, here point \(P( – 7,\,6)\) lies in the second quadrant.
3. In the third quadrant, both \(x\) and \(y\) coordinates are negative so, here point \(R( – 6,\, – 7)\) lies in the third quadrant.
4. In the fourth quadrant, \(x\) coordinate is positive, and \(y\) coordinate is negative so, here Point \(Q(7,\, – 2)\) lies in the fourth quadrant.

Q.5. In the figure given below, what is the perpendicular distance of a point \(P\) from the \(X – axis\).

Ans: To find perpendicular distance from the given point to \(X – axis\) draw a straight line from the point to \(X – axis\) in such a way that the line makes a right angle with \(X – axis\).

Then, the perpendicular distance from a point \(P\) to \(X – axis\) is \(5\,{\rm{units}}\).

Summary

In this article, we learnt about, Cartesian coordinates, Cartesian coordinate system, rectangular coordinate system, Cartesian equation, Cartesian to polar coordinates, polar coordinates to Cartesian, Cartesian plane graph, Cartesian equation of a plane, solved examples on the Cartesian system. We also solved some questions on the Cartesian system to understand the concept.

The learning outcome of this article is how to locate the position of an object on the surface.

Frequently Asked Questions (FAQs) – Cartesian System

Q.1. What is the Cartesian coordinate system?
Ans: A Cartesian coordinate system is a method used to show a point specified in a plane by a set of numerical coordinates.

Q.2. What is the Cartesian system for class \(10\)?
Ans: The system used for describing the position of a point in a plane is also known as the Cartesian system.
The rectangular coordinate system is made up of two integer number lines that intersect at a right angle. The horizontal number line is called the \(x – axis\) and the vertical number line is
called the \(y – axis\). These two-number lines define a smooth surface called a plane, and every point on this plane has been linked to an ordered pair of real numbers \((x,y)\). The first number is called the \(x – \) coordinate, and the second number is called the \(y – \)coordinate.

Q.3. How did the cartesian coordinate system come into existence?
Ans: The Cartesian plane is named after the French mathematician and philosopher René Descartes. He introduced the coordinate system to show how algebra could solve geometric problems.

Q.4. What is a Cartesian equation example?
Ans: Any curve can be represented by having an equation in \(x\) and \(y\) expressing relation between \(x\) and \(y\) coordinates at each point of the curve. Such equations are called Cartesian forms of equations.
For example \({x^2} + {y^2} = 9\)

Q.5. What is the use of the Cartesian plane in real life?
Ans: The Cartesian coordinate plane of \(x\) and \(y\) works well with many simple real-life situations. For instance, if you plan to place different pieces of furniture in a room, you can draw a two-dimensional grid representing the room and use an appropriate unit of measurement.

Q.6. How to define a circle in a cartesian system?
Ans: Cartesian coordinate system with a circle of radius \(2\) centred at the origin is marked in red. The equation of a circle is \({(x – a)^2} + {(y – b)^2} = {r^2}\) where \(a\) and \(b\) are the coordinates of the centre \((a,\,b)\) and \(r\) is the radius.

Example: