Chapter 12.2 - Another Method to Find Coordinates of A Point In Space

In the previous section, we accomplished two tasks:

Task 1:
To find the coordinates of a point with reference to x, y and z-axes.
Task 2: To mark a point in space when the coordinates of that point are given.
(Task 2 is the converse of task 1)
• The same two tasks can be accomplished using another method. In this section, we will see this new method.

◼ First we will see task 1:
• In fig.12.7 below, P is a point in space.

Fig.12.7

• We want to write the position of P. It can be done in 4 steps:
1. The first step always is to draw a plane.
• This plane should be:
    ♦ Passing through P.
    ♦ Parallel to the YZ-plane.
• In the fig.12.7 above, this plane is shown in gray color.
• Mark the point A where this plane intersects the x-axis.
    ♦ Measure the length OA.
• This length is indicated as ‘x’.
    ♦ If A is on the +ve side of the x-axis, then x is +ve.  
    ♦ If A is on the -ve side of the x-axis, then x is -ve.
2. The second step always is to draw a second plane.
• This plane should be:
    ♦ Passing through P.
    ♦ Parallel to the XZ-plane.
• In the fig.12.7 above, this plane is shown in yellow color.
• Mark the point B where this plane intersects the y-axis.
    ♦ Measure the length OB.
• This length is indicated as ‘y’.
    ♦ If B is on the +ve side of the y-axis, then y is +ve.  
    ♦ If B is on the -ve side of the y-axis, then y is -ve.
3. The third step always is to draw a third plane.
• This plane should be:
    ♦ Passing through P.
    ♦ Parallel to the XY-plane.
• In the fig.12.7 above, this plane is shown in magenta color.
• Mark the point C where this plane intersects the z-axis.
    ♦ Measure the length OC.
• This length is indicated as ‘z’.
    ♦ If C is above the XY-plane, then z is +ve.  
    ♦ If C is below the XY-plane, then z is -ve.
4. The fourth and final step always is to write x, y and z together as an ordered triplet: (x,y,z)
• To specify the position of point P, we write: P(x,y,z)

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Let us see two examples.
Example 1:
This can be written in 5 steps:
1. In fig.12.8 below, P is a point in space. Let us assume that, it's position is not known. Our task is to find it's position with reference to the three axes.

Fig.12.8

2. First we draw a plane.
• This plane is:
    ♦ Passing through P.
    ♦ Parallel to the YZ-plane.
• In the fig.12.8 above, this plane is shown in white color.
    ♦ This plane intersects the x-axis at A
    ♦ Length OA is 3 units.
    ♦ A is on the +ve side of the x-axis. So x = 3
3. Next we draw another plane.
• This plane is:
    ♦ Passing through P.
    ♦ Parallel to the XZ-plane.
• In the fig.12.8 above, this plane is shown in yellow color.
    ♦ This plane intersects the y-axis at B.
    ♦ Length OB is 4 units.
    ♦ B is on the -ve side of the y-axis. So y = -4
4. Next we draw a third plane.
• This plane is:
    ♦ Passing through P.
    ♦ Parallel to the XY-plane.
• In the fig.12.8 above, this plane is shown in magenta color.
    ♦ This plane intersects the z-axis at C.
    ♦ Length OC is 3 units.
    ♦ C is on the +ve side of the z-axis. So z = 3
5. So we can write the coordinates of P as: P(3,-4,3)

Example 2:
This can be written in 5 steps:
1. In fig.12.9 below, P is a point in space. Let us assume that, it's position is not known. Our task is to find it's position with reference to the three axes.

Fig.12.9

2. First we draw a plane.
• This plane is:
    ♦ Passing through P.
    ♦ Parallel to the YZ-plane.
• In the fig.12.9 above, this plane is shown in gray color.
    ♦ This plane intersects the x-axis at A
    ♦ Length OA is 4 units.
    ♦ A is on the -ve side of the x-axis. So x = -4
3. Next we draw another plane.
• This plane is:
    ♦ Passing through P.
    ♦ Parallel to the XZ-plane.
• In the fig.12.9 above, this plane is shown in yellow color.
    ♦ This plane intersects the y-axis at B.
    ♦ Length OB is 3 units.
    ♦ B is on the +ve side of the y-axis. So y = 3
4. Next we draw a third plane.
• This plane is:
    ♦ Passing through P.
    ♦ Parallel to the XY-plane.
• In the fig.12.9 above, this plane is shown in magenta color.
    ♦ This plane intersects the z-axis at C.
    ♦ Length OC is 5 units.
    ♦ C is on the -ve side of the z-axis. So z = -5
5. So we can write the coordinates of P as: P(-4,3,-5)

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◼ Now we will see how task 2 is accomplished using this method.
• Consider fig.12.7 that we saw at the beginning of this section. For convenience, it is shown again below:

Fig.12.7

• We are given the coordinates (x,y,z) of point P. We want to mark that point P in space. It can be done in 4 steps:
1. The first step always is to start from O and move along the x-axis.
   ♦ If the given ‘x’ is +ve, we must move in the +ve x direction.
   ♦ If the given ‘x’ is -ve, we must move in the -ve x direction.
• Mark a point A in such a way that, OA = x units.
• Now draw a plane. This plane should be:
    ♦ Passing through A.
    ♦ Parallel to the YZ-plane.
• In the fig.12.7 above, this plane is shown in gray color.
2. The second step always is to start from O and move along the y-axis.
   ♦ If the given ‘y’ is +ve, we must move in the +ve y direction.
   ♦ If the given ‘y’ is -ve, we must move in the -ve y direction.
• Mark a point B in such a way that, OB = x units.
• Now draw a plane. This plane should be:
    ♦ Passing through B.
    ♦ Parallel to the XZ-plane.
• In the fig.12.7 above, this plane is shown in yellow color.
3. The third step always is to start from O and move along the z-axis.
   ♦ If the given ‘z’ is +ve, we must move in the +ve z direction.
   ♦ If the given ‘z’ is -ve, we must move in the -ve z direction.
• Mark a point C in such a way that, OA = x units.
• Now draw a plane. This plane should be:
    ♦ Passing through C.
    ♦ Parallel to the XY-plane.
• In the fig.12.7 above, this plane is shown in magenta color.
4. The point of intersection of the three planes is the position of P.

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Let us see two examples.
Example 1:
This can be written in 5 steps:
1. Consider the fig.12.8 that we saw earlier. For convenience, it is shown again below:

Fig.12.8

2. First we start from O and move along the x-axis.
• Since the given x is +ve, we move along the +ve direction of x-axis.
• We mark the point A in such a way that, length of OA is 3 units.
• Now draw a plane. This plane should be:
    ♦ Passing through A.
    ♦ Parallel to the YZ-plane.
• In the fig.12.8 above, this plane is shown in white color.
3. Next we start from O and move along the y-axis.
• Since the given y is -ve, we move along the -ve direction of y-axis.
• We mark the point B in such a way that, length of OB is 4 units.
• Now draw a plane. This plane should be:
    ♦ Passing through B.
    ♦ Parallel to the XZ-plane.
• In the fig.12.8 above, this plane is shown in yellow color.
4. Next we start from O and move along the z-axis.
• Since the given z is +ve, we move along the +ve direction of z-axis.
• We mark the point C in such a way that, length of OC is 3 units.
• Now draw a plane. This plane should be:
    ♦ Passing through C.
    ♦ Parallel to the XY-plane.
• In the fig.12.8 above, this plane is shown in magenta color.
5. The point of intersection of the three planes is the position of P.       

Example 2:
This can be written in 4 steps:
1. Consider the fig.12.9 that we saw earlier. For convenience, it is shown again below:

Fig.12.9

2. First we start from O and move along the x-axis.
• Since the given x is -ve, we move along the -ve direction of x-axis.
• We mark the point A in such a way that, length of OA is 4 units.
• Now draw a plane. This plane should be:
    ♦ Passing through A.
    ♦ Parallel to the YZ-plane.
• In the fig.12.9 above, this plane is shown in gray color.
3. Next we start from O and move along the y-axis.
• Since the given y is +ve, we move along the +ve direction of y-axis.
• We mark the point B in such a way that, length of OB is 3 units.
• Now draw a plane. This plane should be:
    ♦ Passing through B.
    ♦ Parallel to the XZ-plane.
• In the fig.12.9 above, this plane is shown in yellow color.
4. Next we start from O and move along the z-axis.
• Since the given z is -ve, we move along the -ve direction of z-axis.
• We mark the point C in such a way that, length of OC is 5 units.
• Now draw a plane. This plane should be:
    ♦ Passing through C.
    ♦ Parallel to the XY-plane.
• In the fig.12.9 above, this plane is shown in magenta color.
5. The point of intersection of the three planes is the position of P.

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From the above discussion, we can write three important points:
1. Consider any point P(x,y,z) in space.
    ♦ x will be the distance of P from the YZ-plane.
    ♦ y will be the distance of P from the XZ-plane.
    ♦ z will be the distance of P from the XY-plane.
2. If P is situated on any one of the three axes, then two of it’s coordinates will be zeroes.
    ♦ If P is on the x-axis, then it’s coordinates will be in the form P(x,0,0).
    ♦ If P is on the y-axis, then it’s coordinates will be in the form P(0,y,0).
    ♦ If P is on the z-axis, then it’s coordinates will be in the form P(0,0,z). 
3. If P is situated on any one of the three planes, then one of it’s coordinates will be zero.
    ♦ If P is on the XY-plane, then it’s coordinates will be in the form P(x,y,0).
    ♦ If P is on the XZ-plane, then it’s coordinates will be in the form P(x,0,z).
    ♦ If P is on the YZ-plane, then it’s coordinates will be in the form P(0,0,z).

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• Using the signs of the three coordinates of a point P, we can easily write the octant in which P is situated. We mentioned this in the previous section.
• The converse is also possible. That is., if we are given any of the eight octants, we can easily write the signs of the coordinates of points in that octant.
• Let us see two examples:
Example 1:
What are the signs of the coordinates of points in octant II?
Solution:
1. Octant II is one of the first four octants.
    ♦ So we directly consider quadrant II.
    ♦ Name of quadrant II is X’OY
2. Since octant II is one of the first four octants, we can put Z.
    ♦ So the name of the octant II is X`OYZ
3. Based on this name, the signs are: -ve, +ve, +ve

Example 2:
What are the signs of the coordinates of points in octant VII?
Solution:
1. Octant VII is one of the last four octants.
• The octant occurring above octant VII is: (7-4) = octant III
    ♦ So we consider quadrant III.
    ♦ Name of quadrant III is X’OY’
2. Since octant VII is one of the last four octants, we can put Z’.
    ♦ So the name of the octant VII is X`OY’Z’
3. Based on this name, the signs are: -ve, -ve, -ve

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Now we will see some solved examples. 

Solved example 12.1
In fig.12.10 below, the coordinates of P are (2,4,5). What are the coordinates of the point F ?

Fig.12.10

Solution:
1. The gray plane is parallel to the YZ-plane. So all points on the gray plane will have the same x-coordinate.
• P(2,4,5) lies on the gray plane. So all points on the gray plane will have the x-coordinate ‘2’
• The point F lies on the gray plane. So the x-coordinate of F is ‘2’.
2. The magenta plane is parallel to the XY-plane. So all points on the magenta plane will have the same z-coordinate.
• P(2,4,5) lies on the magenta plane. So all points on the magenta plane will have the z-coordinate ‘5’.
• The point F lies on the magenta plane. So the z-coordinate of F is ‘5’  
3. The point F lies on the XZ-plane.
• All points on the XZ-plane will have the y-coordinate as zero.
4. Based on the above three steps, we get the coordinates of F as: (2,0,5)

Solved example 12.2
Find the octants in which the points (-3,1,2) and (-3,1,-2) lie.
Solution:
Case 1: (-3,1,2)
1. The signs are: -ve, +ve, +ve
    ♦ Based on the first two signs, we can write: X’OY
    ♦ Based on the last sign, we get: Z
• So the name of the octant is X’OYZ
• Since Z is +ve, it is one of the first four octants.
2. The first three letters of the name are: X’OY.
• This corresponds to the quadrant II
3. Since it is one of the first four octants, the names of quadrants and octants are same.
• We get: octant II
Case 2: (-3,1,-2)
1. The signs are: -ve, +ve, -ve
    ♦ Based on the first two signs, we can write: X’OY
    ♦ Based on the last sign, we get: Z’
• So the name of the octant is X’OYZ’
• Since Z is -ve, it is one of the last four octants.
2. The first three letters of the name are: X’OY.
• This corresponds to the quadrant II
3. Since it is one of the last four octants, we must add 4.
    ♦ 2+4 = 6
• We get: octant VI

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The link below gives a few more solved examples:

Exercise 12.1

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In the next section, we will see distance between two points. 

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Chapter 12 - Three Dimensional Geometry

In the previous section, we completed a discussion on conic sections. In this chapter, we will see three dimensional geometry.

The importance of three dimensional geometry can be explained using two simple examples.
Example 1:
This can be written in 5 steps:
1. Consider the point marked as ‘O’ in fig.12.1 below:

Fig.12.1

• Point 'O' is the bottom corner of a large room.
• Point 'O' is formed by the intersection of three items:
   ♦ The green wall.
   ♦ The red wall.
   ♦ The blue floor.
2. The green and red walls are adjacent walls. They are perpendicular to each other. Both walls are perpendicular to the floor.
• So we can write:
The walls and the floor are mutually perpendicular to each other.
3. The electrician wants to fix a lamp.
   ♦ The lamp must be on the green wall.
   ♦ The lamp must be 4 meters away from the red wall.
   ♦ The lamp must be at a height of 3 meters above the floor.
4. To fix the position of the lamp, the electrician adopts two steps:
(i) First he marks point A
• This point is at a distance of 4 meters from O
• The distance 4 meters is measured along a line perpendicular to red wall.
(ii) Then he marks point B
• This point is at a distance of 3 meters from A.
• The distance 3 m is measured along a line perpendicular to the floor.
5. We can write:
• To fix the position of the lamp, the electrician requires the two numbers: ‘4’ and ‘3’.
• He also requires two adjacent perpendicular planes: ‘red wall’ and ‘blue floor’.
6. We are already familiar with this type of problems. The numbers ‘3’ and ‘4’ are the coordinates. They are written as an ordered pair (4,3).
• When viewed from the side of the room, the floor will appear as a line. It is the x-axis.     
• When viewed from the side of the room, the red wall will appear as a line. It is the y-axis.
• When viewed from the side of the room, the green wall will appear as a plane. It is the xy-plane.
• So this is a two-dimensional problem. Every point in a two-dimensional problem will lie in the xy-plane.

Example 2:
This can be written in steps:
1. Fig.12.2 below shows the same room that we saw in example 1.

Fig.12.2

2. The electrician wants to fix a lamp.
   ♦ The lamp must hang from the ceiling.
   ♦ The lamp must be 4 meters away from the red wall.
   ♦ The lamp must be 5 meters away from the green wall.
   ♦ The lamp must be at a height of 3 meters above the floor.
3. To fix the position of the lamp, the electrician adopts three steps:
(i) First he marks point A.
• This point is at a distance of 4 meters from O.
• The distance 4 meters is measured along a line perpendicular to red wall.
(ii) Then he marks point B.
• This point is at a distance of 5 meters from A.
• The distance 5 m is measured along a line perpendicular to the green wall.
(iii) Then he marks point C.
• This point is at a distance of 3 meters from B.
• The distance 3 m is measured along a line perpendicular to the
floor.
4. We can write:
To fix the position of the lamp, the electrician requires the three numbers '4', ‘5’ and ‘3’. He also requires three adjacent perpendicular planes ‘red wall’, ‘green wall’ and ‘blue floor’
5. This is a new type of problem for us. The numbers ‘4’, ‘5’ and ‘3’ are the coordinates. They are written as an ordered triplet (4, 5, 3).
• This is a three-dimensional problem.

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Let us compare the two examples. The comparison can be written in 2 steps:
1. In the example 1, two coordinates and two adjacent perpendicular planes are sufficient to fix the position of an object.
• The two numbers are called coordinates of the object with reference to the two perpendicular planes.
• The object will lie on a third plane which is perpendicular to the first two planes.
• It is a two-dimensional problem. Since we have enough practice in two-dimensional problems, we can ignore the planes and use just the x and y axes.
2. In the example 2, three coordinates and three adjacent perpendicular planes are required to fix the position of an object.
• The three numbers are called coordinates of the object with reference to the three perpendicular planes.
• The object will not be on any of the three planes. We say that, the object is situated in ‘space’.
• It is a three-dimensional problem.

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In many scientific and engineering problems, we will need three coordinates to describe the position of an object. So we must develop a three-dimensional mathematical model which can be easily set up for any given problem. Such a model can be described in 10 steps:

1. Fig.12.3 below shows three mutually perpendicular planes.

Fig.12.3

   ♦ One plane is red in color.
   ♦ Another plane is green in color.
   ♦ The third plane is blue in color.
2. Intersection of the three planes gives us three different lines:
   ♦ Intersection of green and blue planes give us the line X`OX.
         ✰ This line is shown in red color.
   ♦ Intersection of red and blue planes give us the line Y`OY.
         ✰ This line is shown in green color.
   ♦ Intersection of red and green planes give us the line Z`OZ.
         ✰ This line is shown in blue color.
• The three lines obtained in this way are mutually perpendicular to each other. The three lines intersect at O.
3. Now we have two items:
(i) The planes mentioned in step (1)
(ii) The lines mentioned in step (2)  
• These two items together gives us a simple but powerful system to fix the position of any point in space.
• This system is known as the rectangular coordinate system.
• We will soon see how this system can be used to fix points in space.
4. Before that, we will give specific names to each of the different components of the system.
(i) Naming the lines:
   ♦ Line X’OX is called x-axis.
   ♦ Line Y’OY is called y-axis.
   ♦ Line Z’OZ is called z-axis.
(ii) Naming the planes:
• Two axes lie on the blue plane: x-axis and y-axis.
   ♦ This plane is called XY-plane.
   ♦ z-axis is perpendicular to the XY-plane.
• Two axes lie on the green plane: x-axis and z-axis.
   ♦ This plane is called XZ-plane.
   ♦ y-axis is perpendicular to the XZ-plane.
• Two axes lie on the red plane: y-axis and z-axis.
   ♦ This plane is called YZ-plane.
   ♦ x-axis is perpendicular to the YZ-plane.
(iii) The point O is called origin.
5. The blue plane (XY-plane) is considered to be horizontal.
• So Z’OZ (z-axis) will be vertical.
6. XY-plane is considered to be the plane of the paper.
7. Positive and negative distances:
This can be explained in 3 steps.
(i) The XY-plane divides z-axis into two parts: OZ and OZ’.
• Consider the distances measured along the z-axis.
   ♦ Distances measured from O towards Z are taken as +ve.
   ♦ Distances measured from O towards Z’ are taken as -ve.
(ii) The XZ-plane divides y-axis into two parts: OY and OY’.
• Consider the distances measured along the y-axis.
   ♦ Distances measured from O towards Y are taken as +ve.
   ♦ Distances measured from O towards Y’ are taken as -ve.
(iii) The YZ-plane divides x-axis into two parts: OX and OX’.
• Consider the distances measured along the x-axis.
   ♦ Distances measured from O towards X are taken as +ve.
   ♦ Distances measured from O towards X’ are taken as -ve.
8. Naming of octants.
This can be written in 2 steps:
(i) We know that, the XY-plane is horizontal.
• We have a space above this horizontal plane.
   ♦ YZ-plane and XZ-plane together, divides this space into four compartments.
   ♦ Each of these compartments is called an octant.
• The octant formed by +ve x and +ve y axes is called octant I.
   ♦ It can be named as XOYZ.
         ✰ In two-dimensional problems, +ve x and +ve y axes give quadrant I.
         ✰ It is named XOY.
• The octant formed by -ve x and +ve y axes is called octant II.
   ♦ It can be named as X'OYZ.
         ✰ In two-dimensional problems, -ve x and +ve y axes give quadrant II.
         ✰ It is named X'OY.
• The octant formed by -ve x and -ve y axes is called octant III.
   ♦ It can be named as X'OY'Z.
         ✰ In two-dimensional problems, -ve x and -ve y axes give quadrant III.
         ✰ It is named X'OY'.
• The octant formed by +ve x and -ve y axes is called octant IV.
   ♦ It can be named as XOY`Z.
         ✰ In two-dimensional problems, +ve x and -ve y axes give quadrant IV.
         ✰ It is named XOY'.
(ii) We know that, the XY-plane is horizontal.
• We have a space below this horizontal plane.
   ♦ YZ-plane and XZ-plane together, divides this space into four compartments.
   ♦ Each of these compartments is called an octant.
• The octant formed by +ve x and +ve y axes is called octant V.
   ♦ It can be named as XOYZ'.
• The octant formed by -ve x and +ve y axes is called octant VI.
   ♦ It can be named as X`OYZ'.
• The octant formed by -ve x and -ve y axes is called octant VII.
   ♦ It can be named as X`OY`Z'.
• The octant formed by +ve x and -ve y axes is called octant VIII.
   ♦ It can be named as XOY`Z'.
9. We see a pattern in the names of the octants. It can be explained in 5 steps:
(i) Consider octant I. Adding four, we get octant V.
• Both have the same name except the ‘Z’ part.
   ♦ Name of octant I is XOYZ.
   ♦ Name of octant V is XOYZ’.
         ✰ Obviously, V is directly below I.
(ii) Consider octant II. Adding four, we get octant VI.
• Both have the same name except the ‘Z’ part.
   ♦ Name of octant II is X'OYZ.
   ♦ Name of octant V is X'OYZ’.
         ✰ Obviously, VI is directly below II.
(iii) Consider octant III. Adding four, we get octant VII.
• Both have the same name except the ‘Z’ part.
   ♦ Name of octant III is X'OY'Z.
   ♦ Name of octant VII is X'OY'Z’.
         ✰ Obviously, VII is directly below III.
(iv) Consider octant IV. Adding four, we get octant VIII.
• Both have the same name except the ‘Z’ part.
   ♦ Name of octant IV is XOY'Z.
   ♦ Name of octant VIII is XOY'Z’.
         ✰ Obviously, VIII is directly below IV.
(v) So, if we can remember the details about the first four octants, we can easily write the details about the remaining four octants.
• The good news is that, there is not much to remember about the first four octants.
• This is because:
   ♦ the names of the first four octants
   ♦ is similar to
   ♦ the names of the four quadrants in two-dimensional problems.
• This is shown below:
   ♦ Quadrant I is XOY;       Octant I is XOYZ
   ♦ Quadrant II is X'OY;     Octant II is X'OYZ
   ♦ Quadrant III is X'OY';   Octant III is X'OY'Z
   ♦ Quadrant IV is XOY';    Octant IV is XOY'Z
◼ So the bottom line is this:
   ♦ Since we are familiar with the quadrants of two-dimensional problems,
   ♦ we can easily write the details about
   ♦ the eight octants of three-dimensional problems.
• This will become clear when we do actual problems.
10. A comparison between quadrants and octants can be written in 2 steps:
(i) In two-dimensional problems, the x-axis and y-axis together, divides an area into four quadrants. Recall that, 'quad' indicates four.
(ii) In three-dimensional problems, the XY-plane, XZ-plane and YZ-plane together, divides a space into eight octants. Recall that, 'oct' indicates eight.

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Now we know the basic details about the rectangular coordinate system. In the next section, we will see how this system can be used to fix the position of any object in space.

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