Coordinates: Coordinates are two numbers that locate a specific point on a grid, known as a coordinate plane. We know that a point has unique coordinates in a plane. The location of any point on a plane is expressed by an ordered pair of values \((x,\,y)\) and these pairs are known as the coordinates. We use coordinates in our daily life, such as in-class seating, map geographical locations, etc.

In this article, we will discuss the definition of coordinates, what we call the abscissa and ordinate in the coordinates, how to read or plot a point graphically, what are the types of coordinate systems and what is a \(3 – D\) coordinate system and how to represent a point in a \(3 – D\) plane.

Ordered Pair

A pair of numbers \(a\) and \(b\) listed in a specific order with \(a\) at the first place and \(b\) at the second place is called an ordered pair \((a,\,b)\).
Note that \((a,\,b) \ne (b,\,a)\)
Thus, \((2,\,3)\) is one ordered pair and \((3,\,2)\) is another ordered pair.

Types of Coordinate System

We have mainly two types of coordinate systems:

Cartesian Coordinate System

To locate the position of a point in a plane we use the cartesian coordinate system. It uses the concept of mutually perpendicular axes to denote the coordinate of a point. Points are represented in the form of ordered pair of \(x\) and \(y\)-coordinates represented as \((x,\,y)\) in the two-dimensional plane.

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.

Polar Coordinate System

The polar coordinate system is a different way to express points in a plane. Basically, we have two parameters, namely angle and radius. The angle \(\theta \) with the polar axis has a single line through the pole measured anti-clockwise from the axis to the line. The point will have a unique distance from the origin \(r\). Thus, a point in the polar coordinate system is represented as a pair of coordinates \((r,\,\theta )\).
The distance from the pole is called the radial distance or simply radius and the angular coordinate, polar angle.
Consider the figure below that shows the relationship between polar and cartesian coordinates.

\(x = r\,\cos \,\theta \) and \(y = r\,\sin \,\theta \)

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

The polar equation of a curve consists of points of the form \((r,\,\theta )\).

Coordinate Axes

The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines, called the coordinate axes. They are marked as the \(X\)-axis or the horizontal axis and the \(Y\)-axis or the vertical axis.

Coordinates

The location of any point on a plane is expressed by an ordered pair \((x,\,y)\) and these pairs are known as the coordinates.
The point at which both the coordinate axes intersect is called the origin, \(O(0,\,0)\).

Sign Convention in Cartesian Plane

A Cartesian plane is divided into four quadrants by two coordinate axes perpendicular to each other.
The four quadrants, along with their respective sign convention of ordered pairs, are represented in the graph below.
1. If a point is in the \({\rm{I}}\) quadrant, then the ordered pair will be in the form \({\rm{( + , + )}}\).
2. If a point is in the \({\rm{II}}\) quadrant, then the ordered pair will be in the form \({\rm{( – , + )}}\).
3. If a point is in the \({\rm{III}}\) quadrant, then the ordered pair will be in the form \({\rm{( – , – )}}\).
4. If a point is in the \({\rm{IV}}\) quadrant, then the ordered pair will be in the form \({\rm{( + , – )}}\).

The figure given below shows the coordinates \({\rm{(2, 3)}}\) lies in the \({\rm{I}}\) quadrant since both the coordinates are positive.

The \(x\)-coordinate of a point is its perpendicular distance from the \(X\)-axis measured along the \(X\)-axis (positive along the positive direction of the \(X\)-axis and negative along the negative direction of the \(X\)-axis). The \(x\)-coordinate is also called the abscissa.

For the point \(P\), the \(x\)-coordinate is \(+4\) and for \(Q\), it is \(-6\).
The \(y\)- coordinate of a point is its perpendicular distance from the \(X\)-axis measured along the \(Y\)-axis (positive along the positive direction of the \(Y\)-axis and negative along the negative direction of the \(Y\)-axis). The \(y\)-coordinate is also called the ordinate.
For the point \(P\), the \(y\)-coordinate is \(+3\) and for \(Q\), it is \(-2\).

Reading or Plotting Coordinates in Graph

We always write coordinates in brackets, the two coordinates are separated by a comma. As coordinates are ordered pairs of numbers; the first number represents the point on the \(X\)-axis and the second represents the point on the \(Y\)-axis.
When reading or plotting coordinates, we always go across first and then up. A good way to remember this is: Across the landing and up the stairs. To plot the points \({\rm{(4, 5)}}\) in the Cartesian coordinate plane, we follow the \(X\)-axis until we reach \(4\) and draw a vertical line at \(x = 4\).
Similarly, we follow the \(Y\)-axis until we reach \(5\) and draw a horizontal line at \(y = 5\). The intersection of these two lines is the position of \({\rm{(4, 5)}}\) in the Cartesian plane. This point is at a distance of \(4\) units from the \(Y\)-axis and \(5\) units from the \(X\)-axis. Thus, the position of \({\rm{(4, 5)}}\)) is located in the cartesian plane.

3-D Coordinates

A \(3\)-dimensional Cartesian coordinate system is formed by a point called the origin (denoted by \(O\)) and a basis consisting of three mutually perpendicular vectors. These vectors define the three coordinate axes. They are also known as the abscissa, ordinate and applicate axis, respectively. The coordinates of any point in space are determined by three real numbers: \(x,\,y,\,z\).
We represent the point \(\) by the ordered triplet \((a,\,b,\,c)\) of real numbers, and we call \(a,\,b\) and \(c\) the coordinates of \(P\); \(a\) is the \(x\)-coordinate, \(b\) is the \(y\)-coordinate, and \(c\) is the \(z\)-coordinate.
Thus, to locate the point \((a,\,b,\,c)\), we can start at the origin \(O\) and move \(a\) units along the \(X\)-axis, then \(b\) units parallel to the \(Y\)-axis, and then \(c\) units parallel to the \(Z\) -axis as shown in figure.

Solved Examples – Coordinates

Q.1. Plot the following points \(A\left( {2,2} \right),\,B\left( { – 2,2} \right),C\left( { – 2, – 1} \right),\,D\left( {2, – 1} \right)\) in the Cartesian plane. Discuss the type of the diagram by joining all the points taken in order.
Ans: We need to plot the following points \(A(2,\,2),\,B( – 2,\,2),\,C( – 2,\, – 1),\,D(2,\, – 1)\). First, check the quadrants where to locate these points.

\(ABCD\) is a rectangle.

Q.2. In which quadrant do the following points lie?
(a) \((3,\, – 8)\)
(b) \(( – 1,\, – 3)\)
(c) \((2,\,5)\)
(d) \(( – 7,\,3)\)
Ans: (a) The \(x\)- coordinate is positive, and \(y\) – coordinate is negative. So, point \((3,\, – 8)\) lies in the \({\rm{IV}}\) quadrant.
(b) The \(x\)-coordinate is negative, and \(y\) – coordinate is negative. So, point \(( – 1,\, – 3)\) lies in the \({\rm{III}}\) quadrant.
(c) The \(x\)-coordinate is positive, and \(y\) – coordinate is positive. So point \((2,\,5)\) lies in the \({\rm{I}}\) quadrant.
(d) The \(x\)-coordinate is negative, and \(y\) – coordinate is positive. So, point \(( – 7,\,3)\) lies in the \({\rm{II}}\) quadrant.

Q.3. Write the coordinates of the points marked on the axes in the figure.

Ans:
1. The point \(A\) is at a distance of \(+4\) units from the \(Y\)-axis and at a distance zero from the \(X\)-axis. Therefore, the \(x\)-coordinate of \(a\) is \(4\) and the \(y\)-coordinate is \(0\). Hence, the coordinates of \(A\) are \((4,\,0)\).
2. The coordinates of \(B\) are \((0,3)\).
3. The coordinates of \(C\) are \((-5,0)\).
4. The coordinates of \(D\) are \((0,-4)\).
5. The coordinates of \(E\) are \(\left( {\frac{2}{3},\,0} \right)\).

Q.4. See the figure given below and complete the following statements:
1. The abscissa and the ordinate of the point \(B\) are __ and __, respectively. Hence, the coordinates of \(B\) are __,__.
2. The \(x\)-coordinate and the \(y\)-coordinate of the point \(M\) are __ and __, respectively. Hence, the coordinates of \(M\) are __,__.
3. The \(x\)-coordinate and the \(y\)-coordinate of the point \(L\) are __ and __, respectively. Hence, the coordinates of \(L\) are __,__.
4. The \(x\)-coordinate and the \(y\)-coordinate of the point \(S\) are __ and __, respectively. Hence, the coordinates of \(x\) are __,__.

Ans:
1. Since the distance of the point \(B\) from the \(Y\)-axis is \(4\) units, the \(x\)-coordinate or abscissa of the point \(B\) is \(4\). On the other hand, the distance of the point \(B\) from the \(X\)-axis is \(3\) units; therefore, the \(Y\)-coordinate, i.e., the ordinate, of the point \(B\) is \(3\). Hence, the coordinates of the point \(B\) are \(\left( {4,\,3} \right)\).
2. The \(x\)-coordinate and the \(y\)-coordinate of the point \(M\) are \(-3\) and \(4\), respectively. Hence, the coordinates of the point \(M\) are \(\left( {-3,\,4} \right)\).
3. The \(x\)-coordinate and the \(y\)-coordinate of the point \(L\) are \(-5\) and \(-4\), respectively. Hence, the coordinates of the point \(L\) are \(\left( {-5,\,-4} \right)\).
4. The \(x\) – coordinate and the \(y\)- coordinate of the point \(S\) are \(3\) and \(-4\), respectively. Hence, the coordinates of the point \(S\) are \(\left( {3,\,-4} \right)\).

Q.5. Write down the coordinates of the points \(P,\,Q\) and \(R\)

Ans: The point \(P\) is at \(-3\) units from \(X\)-axis, and the \(y\) and \(z\)-coordinates are zero.
So, the coordinates of the point \(P\) is \(( – 3,\,0,\,0)\)
The point \(Q\) is at \(6\) units from \(X\)-axis, \(-2\) units from \(Y\)-axis and \(z\)-coordinates is zero.
So, the coordinates of the point \(Q\) is \((6,\, – 2,\,0)\)
The point \(R\) is at \(-3\) units from \(X\)-axis, \(-2\) units from \(Y\)-axis and \(5\) units from \(Z\)-axis.
So, the coordinates of the point \(R\) is \(( – 3,\, – 2,\,5)\)

Summary

In this article, we have discussed that we use coordinates to locate the point in the plane and it is always written in the ordered pair like we write \(x\)-coordinate in the first position and \(x\)-coordinate in the second position. The abscissa is the \(x\)- coordinate, and the ordinate is the \(y\)-coordinate.
We have also discussed the types of coordinate systems like the cartesian and the polar coordinate system. Also, we have seen how to represent coordinates in \(3\) – dimensional plane.

FAQs – Coordinates

Q.1. What are the coordinates of the origin \(0\)?
Ans: It has zero distance from both the axes so its abscissa and ordinate are both zero. Therefore, the coordinates of the origin are \((0,\,0)\).

Q.2. Why do we need coordinate geometry?
Ans: Coordinate geometry has various applications in real life. Some of them are given below:

1. To locate the position of cursor or finger in digital devices like computers, mobile phones, etc.
2. To determine the position and location of aeroplanes accurately in aviation.
3. In maps and in navigation (GPS)
4. To map geographical locations using latitudes and longitudes.

Q.3. What do you mean by coordinates?
Ans: The location of any point on a plane is expressed by an ordered pair of values \((x,\,y)\) and these pairs are known as the coordinates.

Q.4. How do you write the coordinates of a point?
Ans: We always write coordinates in brackets, with the two numbers separated by comma. Coordinates are ordered pairs of numbers; the first number represents the point on the \(X\)-axis and the second the point on the \(Y\)-axis.

Q.5. What is abscissa and ordinates in coordinate geometry?
Ans: Abscissa is the \(x\)-coordinate and ordinate is the \(y\)-coordinate.

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