In the previous section we succeeded in fixing the position of the lamp. How do we represent this position on a sheet of paper?
The usual A4 size paper that we use has a width of 21 cm and a length of 29 cm. This is shown in the fig.2.6 below:
But we have to plot 40 cm and 90 cm. Obviously, we cannot mark these distances on an A4 size paper. So we use a method known as 'scaling'. It is done as follows:
We assume that each 1 cm that we draw on paper represents 10 cm of the actual measurement. So to represent 40 cm, we need to draw only 4 cm. And to represent 90 cm we need to draw only 9 cm. So the scale of the drawing is 1cm = 10 cm. It is also written as 1:10. In this way, the drawing can be kept within the boundaries of the sheet of paper. Also recall that the measurements were made from two perpendicular walls AB and AD. These walls are the 'references'. Without some kind of reference, we will not be able to start our measurements. So the walls also have to be suitably represented on paper. It is done using two perpendicular 'axes'. The horizontal axis is called the X axis and the vertical one is called the Y axis. An arrow mark is given at the ends of the axes. This is to indicate that the axes can be extended to 'any' suitable distance. The drawing is shown in fig.2.7 below:
The procedure for making such a drawing can be summarized as follows:
• On A fresh A4 size paper, draw the horizontal X axis and the vertical Y axis
• The point of intersection of the two axes is called the origin
• From the origin, mark off 1 cm intervals towards the right on the X axis
• From the origin, mark off 1 cm intervals upwards on the Y axis
• The distance of each mark (from the origin) is to be written near that mark. The distances on the paper are 1 cm, 2 cm, 3 cm . . . and so on.
• But each 1 cm represents 10 cm. So we write 10 cm, 20 cm, 30 cm . . . and so on.
• The distance of the origin from the origin is of course 'zero' in both the X and Y directions. So we write (0,0) at the origin
• If all the distances are written, there will be congestion of space. We need only write alternate distances. So we write 20, 40, 60 . . . and so on.
• Now we are ready to mark the position of the lamp. Draw a vertical dotted line through the 40 cm mark on the X axis. Draw a horizontal dotted line through the 90 cm mark on the Y axis.
• The point of intersection of these two dotted lines represents the position of the lamp.
So we succeeded in representing the position of the lamp on a sheet of paper. But there is another method which will make the above steps, a lot more easier and faster. In that method, we use a Graph paper. The graph paper already has a grid on it. A grid formed by vertical and horizontal lines.
The main features of a graph paper are given below:
The usual A4 size paper that we use has a width of 21 cm and a length of 29 cm. This is shown in the fig.2.6 below:
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| Fig.2.6 Size of an A4 paper |
But we have to plot 40 cm and 90 cm. Obviously, we cannot mark these distances on an A4 size paper. So we use a method known as 'scaling'. It is done as follows:
We assume that each 1 cm that we draw on paper represents 10 cm of the actual measurement. So to represent 40 cm, we need to draw only 4 cm. And to represent 90 cm we need to draw only 9 cm. So the scale of the drawing is 1cm = 10 cm. It is also written as 1:10. In this way, the drawing can be kept within the boundaries of the sheet of paper. Also recall that the measurements were made from two perpendicular walls AB and AD. These walls are the 'references'. Without some kind of reference, we will not be able to start our measurements. So the walls also have to be suitably represented on paper. It is done using two perpendicular 'axes'. The horizontal axis is called the X axis and the vertical one is called the Y axis. An arrow mark is given at the ends of the axes. This is to indicate that the axes can be extended to 'any' suitable distance. The drawing is shown in fig.2.7 below:
 |
| Fig.2.7 Representation of the position of the Lamp |
• On A fresh A4 size paper, draw the horizontal X axis and the vertical Y axis
• The point of intersection of the two axes is called the origin
• From the origin, mark off 1 cm intervals towards the right on the X axis
• From the origin, mark off 1 cm intervals upwards on the Y axis
• The distance of each mark (from the origin) is to be written near that mark. The distances on the paper are 1 cm, 2 cm, 3 cm . . . and so on.
• But each 1 cm represents 10 cm. So we write 10 cm, 20 cm, 30 cm . . . and so on.
• The distance of the origin from the origin is of course 'zero' in both the X and Y directions. So we write (0,0) at the origin
• If all the distances are written, there will be congestion of space. We need only write alternate distances. So we write 20, 40, 60 . . . and so on.
• Now we are ready to mark the position of the lamp. Draw a vertical dotted line through the 40 cm mark on the X axis. Draw a horizontal dotted line through the 90 cm mark on the Y axis.
• The point of intersection of these two dotted lines represents the position of the lamp.
So we succeeded in representing the position of the lamp on a sheet of paper. But there is another method which will make the above steps, a lot more easier and faster. In that method, we use a Graph paper. The graph paper already has a grid on it. A grid formed by vertical and horizontal lines.
The main features of a graph paper are given below:
• All the vertical lines are parallel to each other.
• The distances between all the vertical lines are equal
• All the horizontal lines are parallel to each other
• The distances between all the horizontal lines are equal and is same as the distances between the vertical lines (usually, this distance is equal to 1 cm)
• The vertical and horizontal lines are exactly perpendicular to each other.
• The grid divides the area into 1 cm x 1cm square units.
• Each 1cm is further divided into 10 equal parts by thinner lines. So each subdivision is equal to 1mm. Thus the small squares are 1 mm x1 mm
• On some graph papers, an intermediate subdivision is given at 5mm. This is marked by lines of intermediate thickness.
On a plain sheet of paper, we will have to ensure that 'parallel and perpendicular properties are maintained' in every step, right from the beginning of drawing the two axes. For that we will have to use set squares, protractors etc., But on a graph paper, as the grid lines are exactly parallel and perpendicular, markings can be done quickly, without using special instruments.
The fig.2.8 below shows the representation of the position of the lamp on a graph paper. The position is indicated using an 'x' mark. This point can be easily reached by following the steps given below:
• First draw the X and Y axes. They can be drawn over suitable grid lines. This will ensure that they are perpendicular to each other
• Assume 1 cm represents 10 cm, and mark the intervals on both the axes
• From the origin move 40 cm to the right along the X axis. Thus the '40 cm mark' is reached
• From there move 90 cm upwards. This is the final position
• We can see that the final position is aligned with 90 cm on the Y axis and 40 cm on the X axis
• The grid lines help us to reach this point with out using any special instruments.
• At the final position, it is written: (40,90). Here, 40 and 90 are the coordinates of the point. 40 is the x- coordinate (because we moved 40 cm in the direction of the X axis) and 90 is the y- coordinate (because we moved 90 cm in the direction of the Y axis) . We say: The coordinates of the point are (40,90)
The 17th century mathematician Rene Descartes developed the system of fixing a point with the help of two measurements, vertical and horizontal. This system came to be known as Cartesian system, in his honour.
Now we know the significance of using coordinates for fixing up positions. A similar application is used in the seating arrangement in an auditorium. In an auditorium, the seats will be arranged in rows. If your Ticket says that the seat number is D3, it means that your seat is the third one in the ‘D’ row. This is illustrated in the fig.2.9 below.
In the above fig, the rows are named from bottom to top, and in each row, the seats are numbered from left to right. This may not be the system in all cases. So the seat should be confirmed by looking at the actual seat number. It will be written at the side of each seat.
In the next section we will see some numerical examples.
• The distances between all the vertical lines are equal
• All the horizontal lines are parallel to each other
• The distances between all the horizontal lines are equal and is same as the distances between the vertical lines (usually, this distance is equal to 1 cm)
• The vertical and horizontal lines are exactly perpendicular to each other.
• The grid divides the area into 1 cm x 1cm square units.
• Each 1cm is further divided into 10 equal parts by thinner lines. So each subdivision is equal to 1mm. Thus the small squares are 1 mm x1 mm
• On some graph papers, an intermediate subdivision is given at 5mm. This is marked by lines of intermediate thickness.
On a plain sheet of paper, we will have to ensure that 'parallel and perpendicular properties are maintained' in every step, right from the beginning of drawing the two axes. For that we will have to use set squares, protractors etc., But on a graph paper, as the grid lines are exactly parallel and perpendicular, markings can be done quickly, without using special instruments.
The fig.2.8 below shows the representation of the position of the lamp on a graph paper. The position is indicated using an 'x' mark. This point can be easily reached by following the steps given below:
 |
| Fig.2.8 Position on a Graph paper |
• First draw the X and Y axes. They can be drawn over suitable grid lines. This will ensure that they are perpendicular to each other
• Assume 1 cm represents 10 cm, and mark the intervals on both the axes
• From the origin move 40 cm to the right along the X axis. Thus the '40 cm mark' is reached
• From there move 90 cm upwards. This is the final position
• We can see that the final position is aligned with 90 cm on the Y axis and 40 cm on the X axis
• The grid lines help us to reach this point with out using any special instruments.
• At the final position, it is written: (40,90). Here, 40 and 90 are the coordinates of the point. 40 is the x- coordinate (because we moved 40 cm in the direction of the X axis) and 90 is the y- coordinate (because we moved 90 cm in the direction of the Y axis) . We say: The coordinates of the point are (40,90)
The 17th century mathematician Rene Descartes developed the system of fixing a point with the help of two measurements, vertical and horizontal. This system came to be known as Cartesian system, in his honour.
Now we know the significance of using coordinates for fixing up positions. A similar application is used in the seating arrangement in an auditorium. In an auditorium, the seats will be arranged in rows. If your Ticket says that the seat number is D3, it means that your seat is the third one in the ‘D’ row. This is illustrated in the fig.2.9 below.
 |
Fig.2.9
Seating arrangement using coordinates
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In the next section we will see some numerical examples.
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