UNIT 10 INTRODUCING 3 ~ - G E O M E T ~ Y

Structure Page Nos.

10.1 Introduction
Objectives
10.2 Visualising the Coordinate System
10.3 Change of Axes
Translation
Rotation
10.4 Lines -Skew and Others
10.5 Locus
10.6 Summary
10.7 Comments on Exercises

10.1 INTRODUCTION
\

It isgenerally felt by zt\most all teachers that students find it difficult to visualise
abstract three-dimensiopal(3D) figures when they are discussed in the classroom.
The main reason is that while teaching them concepts in three-dimensional geometry,
we have to draw the figures on the blackboard or on paper, i.e., $2D. Often, these,
representations do not reflect the entire picture, and a lot is left to the imagination.

In Sec. 10.2 of this unit, we begin with suggesting ways of helping your studenls to
visualise 3D from 2D pictures. We have also suggested strategies for helping them to
represent 3D in 2D.

In Sec. 10.3 and Sec. 10.4 we continue this discussion in the context of shifting the
coordinate system. We look at ways in which the experiences of the students in 2D
can be used for helping them to understand rigid body transformations in 3D. We also
consider ways of helping them to visualise pairs of skew lines, and ways of generating
infinitely many such pairs.

Finally, in Sec. 10.5, we take up a major area of difficulty that students face while ,
studying three-dimensional geometry. This is related to solving problems about loci.,
Here, we have suggested some strategies that would clarify this concept in their
minds.

Throughout this unit, you will find that we have tried to use the Cartesian and the
vectorial approaches. This is important because students must rkalise their
equivalence, and when it is convenient to use which approach.

A word about the abbreviations we use! In this unit we shall interchangeably use '2D'
or '2-space' for 'two-dimensional space'. Similarly, '3D' or '3:ipce' will mean
'three-dimensional space'.

Objectives

After studying this unit, you should be able to improve the ability ~ f , ~ o lwerq r s to

visualise a 3D coordinate system;

draw and read 2D representations of 3D objects and situations;
visualise the translation and rotation of axes in 3D, and. sbfve ?elated
visualise skew lines and differentiate between skew'and lp&HC\ lines;
6 explain what 'locus' is, and identify it algebrddly as well as geometrikally in
some simple situatiods.
Geometry and Mechanics
10.2 VISUGLISING THE COORDINATE SYSTEM
While talking to stddents and teachers of Classes 11 and 12, we realised that a major
problem faced by many students is their inability to read or draw 2D representations of
objects in 3D space. This is probably because we teachers assume that the students
will understand the representation of 3D space that we draw on the board for them. As
long as the students are studyinglreadingldrawing in two-dimensional space, there is
no problem. This is because there is not much left to the imagination, for instance, a
circle or disc on paper is the same as it is in reality. But, when they start studying
positions and movement in three-dimensional space, which is represented on paper
(i.e., in 2D), there is a lot of imagining that they need to do. They need to 'see' how
the third dimension (i.e., depth) is added to the diagram, merely by drawing an axis (a
line) on the paper at an acute angle to one of the other axes. This ability of imagining
the third dimension is not acquired easily. For visualising the complete image, a
person needs several opportunities to correlate a variety of concrete models with their
2D representations. It is this kind of exposure that will help her to become familiar
with the conventions of drawing 3D in 2D.

Sometimes, the stqdents may not realise all the conventions that are being used. You
could ensure that this doesn't happen by asking them questions that require them to
focus on these co&entions. You could also draw some concrete situations in which
you deliberately ignore a particular convention, and ask them if it matters, and in what
way. You should also ask students to draw 3D situations or objects on paper. Then
ask each student to clearly explain which conventions she has used for drawing the 2D
representations of the 3D objects and of 3-space. Of course, some teachers say that all
this is not necessary. According to them, the students are already familiar with these
conventions because they see photographs, diagrams, etc., around them all the time -
all of which are 2 0 representations of 3D objects. However, remember that the
student is familiar with these 3D objects, and therefore, would usually not look at
them carefully regarding the way details related to the third dimension are presented.

Let us now look at an example of how the classroom can be used for familiarising the
learner with visualisation of 3D in 2D. Of course, the assumption (which needs to be
checked!) is that the students are familiar with finding coordinates in 2D. With this
assumption, let US consider the following activity.

To start with, hovV is your classroom usually arranged? Are you, the teacher, facing
about 30 students with the board on the wall behind you? Let us assume the room is
as in Fig. l(a).

Fig. 1
 Introducing 3DGeometry
You could ask your students to assume that the corner of the floor to your right, where r

two walls and the floor meet, is the origin. You could mark the x and y axes along the
edges of the floor, and the z-axis upwards, showing the height from the floor. Here, it
is important that the students realise the order in which the 3 axes are taken -
according to the right-hand rule. Give them an example of an order that is not
acceptable, according to convention.

Each student could be askdd to find the coordinates of, say, her left foot, w.r.t. these x 'w.r.t' is often used in
and y axes. Then you could ask her to stand at her place, and take her height as the rnaths, and stands for
'with respect to'.
third dimension, measured along the z-axis (see Fig. l(b)).
\

Next, you cauld draw a scheniatic view of the classroom on the blackboard, showing
the 3 axes and explaining how and why this is an acceptable 2D representation. You
could pinpoint the positions of the heads of one or two students in this diagram by
writing the coordinates next to the point, and showing how those points are placed in
the diagram. The rest of the students could, then, add their relative positions in the
same diagram. This would help them in understanding how the diagram represents
the real 3D situation. However, we must remember that some of the students may
need several hints regarding how to find the 3 coordinates of any point in space.

You need to do several activities of this kind with them. This activity could be
extended by, for instance, asking them to form teams. These teams could ask each
other the coordinates of various objects w.r.t. the axes chosen -the tip of a bulb, the
centre of the fan, the foot of a chair leg, and so on. Then, one of them could mark a
point on your diagram, giving its coordinates, and ask what object in the room
corresponds to it.

With such activities and exercises, the students would be able to relate any given point
P in the room, with the three perpendicular distances of this point from the mutually
perpendicular planes, namely, the planes of the floor and two adjacent walls of the ,
room. They would be able to do the converse too -given any three positive
numbers, they could take these three as the distances measured from the three planes
and locate a corresponding point P in the room.
'fz >
A

I
These 'room-related' activities require to be followed by more abstract activities to
ensure that the students can improve their visualisation. As we've mentioned earlier,
let us look at the problem students can have, for instance in 'reading' Fig. 2. In this \
figure of a cube, L BOC is clearly a right angle. However, L AOB and LAOC are
not right angles. You have to help your students look at Fig.2 in such a manner that
they visualise the edge OA as sticking out of the sheet of paper/bla'ckboard,
perpendicular to the plane of the paperhlackboard. They have to realise &at L AOB c' I
and L AOC, when coming out of the plane, are at right angles to OBA'C. It is then Fig. 2 /
that we have three mutually perpendicular lines OA, OB and OC, and three mutually
perpendicular planes OAC'B, OBA'C and OCB'A. So, if they see this cube as
representing their classroom, then they should see that it shows the opposite edges of
the floor and the ceiling as being parallel. Similarly, can they see that it shows the
floor and the ceiling as parallel planes, and the opposite walls as parallel planes?
'- J,

With the different kinds of experiences suggested above, the students would see that if
the axes were chosen as in Fig. 1, it is adequate to have 3 positive numbers to locate a
point inside the room. Now, ask the students what happens if any of the positive
numbers is greater than the dimension of the room (or the cube in Fig. 2)? Can they
reply that the corresponding point would be outside the room in the direction that this
number corresponds to? And, what happens if you had taken P to be a point in the
verandah behind the wall (i.e., the plane XOZ in Fig. I), or to the left of the plane
YO2 in the conidor, or below the plane (floor) XOY? Do they realise that the
corresponding distances measured would be in the opposite directions, and so the
corresponding coordinates would become negative? It is not easy for the students to
 I
$eometry and Mechanics unklerstand whikh points correspond to negative coordinates along any axis. You
would need to be patient about this aspect, and get them to look at it again and again
through a variety of appropriate activities.

Having discussed these points with your students, you could give themsome simple
exercises which allow them to practise identifying co-ordinates and locating points.
Some are givenlbelow. Try them with your students.

El) i) In Fig. 3, the point P has coordinates (x,y,z). Write down the
coordinates of the points A,B,C, A',B',C' and 0.
ii) Write down the coordinates of the feet of the perpendiculars from the
point (-1,2,-4) to the coordinate planes.
iii) Find the perpendicular distance of the point (-3,-1,4) from the y-axis.
iv) Find the image of the point (-3,2,1) iil the ZX-plane.

G2) Frame some questions to assess if your students are able to distinguish
between'the eight octants. Try them out with your students. What is the
outcome? If the assessment is negative, how would you help your students to
become familiar with the octants?

E3) What kind of detailed activity would you do with your students to familiarise
them with negative numbers as coordinates?
I

In fact, it is with several activities/exercises/focussed questions of this kind that your

students would realise that given any three real numbers rl, r;! and rg, there is one and
I
only one point located in space corresponding to them'in that order, i.e., the point
(r,, r2, r3). Here, you would need to do some exercises with them to focus their .
attention on the fact that (rl, rz, r3)is not the same point as (rl, rz, r3), in general.
Ask them to think about the reason for this, and to give you examples in support of
their reason.

At the next level of abhraction, you would need to help your students realise the
arbitrariness of the cboice of the origin and the axes. This arbitrariness is in se-1
g h same origin, there are infinitely many sets of 3 mutually
ways. For one, t h r ~ ~ the
perpendicular lines. Secondly, anypoint in space can be taken as the origin.
However, we will look at the formal aspect of this arbitrariness in detail in the next
section.

There is yet anotkr source of confusion that students have. This is related to the unit
lengths to be chokn along the 3 axes, Students think h a t it should be the same for all
the axes. This is because in our ex ples we usually choose the unit lengths to be the
3
same. This is regardless of axes in 2 or in 3D. We need to consciously give them
examples that forae a choice of different unit lengths along the axes. These examples
would first need to be related to 2D, which the students are more familiar with.

E4) Write down two problems you would do with students to help them realise
that the uqit distances chosen along the axes need not be the same.

Before ending this section, we should mention a difficulty students have about an
equivalent approach to locate a point in space, the vectorial approach. The students
need to understand the equivalence under which any point P(x,y,z) corresponds to the
vector OP. Helping students to understand and,visualise this equivalence would
require the use of several activities. Some of them would be built around concrete 3D
models. Think abbut other activities while doing the following exercises.
 Introducing 3D-Geometry

E5) How would you help your students correlate the two approaches for pinpointing
any point in space? Does your method help them understand why the vector
. OP is called the position vector of the point P w.r.t. thk origin of reference O?

E6) What is the direction of a vector whose magnitude is zero?

E7) What is the difference between 0 and O? . -.

Students may also wonder about why they need to bother about two approaches, one is'
bad enough! Here is where you could tell them instances of where the Cartesian
approach can be more useful, and those in which the vectorial approach is more
handy. Some examples are alternative representations for a sphere with centre rl, and
radius a, or a plane perpendicular to a given vector and at a distance d &om the origin,
etc. You should think of many other examples.

Let us now focus on an aspect that students usually don't realise - the arbitrariness
of the choice of the origin and axes.

10.3 CHANGE OF AXES

It is important for students to know that there is nothing sacred about the particular
point in a particular plane that is chosen as the origin. After all, the axes do not
actually exist. It is we who create them -picking any point as an origin and picking
any 3 mutually perpendicular lines as the axes. To help students internalise this, you
could first discuss the impact of changing the origin and the axes in two dimensions
only. You could draw pairs of axes so that neither is horizontal, arid ask the students
their opinion on this. Let them underitand why this is acceptable, as long as the
convention is followed that the direction of the y-axis is obtained by rotating the
x-axis through 90" in the anti-clockwise direction.

The students could then be asked to think of ways of generalising this to three
dimensions. We could also ask them : what happens if we just shift the axes? How
are the coordinates of the objects changed by such changes? Do the objects
themselves change?
Let us focus on familiarising our students with two particular kinds of transformations
of the axes here. We take them up in the following sub-sections, one by one.
10.3.1 Translation
Let us start by asking your students to pick any point on the board as an origin 0
(0,0), and draw the axes OX and OY. Next, they should pick any other point 0' on
the board as the new origin. Suppose its coordinates are (a,b) w.r.t the system XOY.
Draw 0' X' and 0' Y' parallel to OX and OY, respectively, and choose them as the
new axes (see Fig. 4). Now you could ask your students what the coordinates of 0
and 0' are w.r.t. X' 0' Y'. Are they able to answer (- a, -b) and (0,0), respectively?
Next, suppose the coordinates of any point P in the two systems are (x,y) and (x', y'),
respectively. Ask your students if these coordinates are related to each other, and
how? You may need to give them -- a hint (as in Fig. 4.)
From the figure, do they see the following relationship?
e
x = X' + a, y = y' + b, or, equivalently, (1)
x'=x-a,y'=y-b (2)
This ,transformation of the axes by shifting the origin is called a translation. In the
Geometry and Mechanics

Fig. 4

example above, we have translated the origin from 0 to Of,keeping the axes parallel
to their original position. I
Now, students m+ty wonder about what the point is of translating the axes. A good
way of showing them the utility is to give them several examples of equations which
are in non-standard form, and which reduce to 'knowh' simple forms by shifting the
origin to an apprapriate point. For instance, if we shift the origin to the centre of a
circle, its equation reduces to standard form. Ana this form is much easier to deal
with geometrically or algebraically. You could give them examples like the equation
x2 + yz - 2x + 4y + 1 = 0.
This represents a Gircle with centre (1, -2), a fact that is not obvious to students. But,
by shifting the origin to (1, -2), and using the equations (2), this equation changes to

a form that students can immediately recognise =.a circle with radius 2 units.
As important as the algebraic translation is the geometric understanding of what is
going on. This is what many students don't understand. And, tkhs is what you need to
explain to them. How are the two objects, before and after translation, related? A
good way is to draq the circle xj2+ y'2 = 4 w.r.t. the X'O'Y' system. Then, keeping
this diagram, delete the X'O'Y' system and bring back the original XOY system. This
way the students see the placing of the circle w.r.t. the original axes.
You should ask them to do this kind of exercise with several objects -lines, conics,
etc. Some such exercises are given below.

E8) Ask your students to simplify the following equations:

i) 2x - 3y - 5 = 0, by shifting the origin to (1, -1).
ii) x2 + yZ - 6x + 4y + 4 = 0, by shifting theorigin to (3, -2).

Also ask them to explain what this means geometrically.

E9) ~ h i the
8 origin in such a manner that the equation
x2 + y2 + 2x - 8y + 16 = 0
/
is in standard,form. What is happening geometrically?
 htroducing 3D-Geometry
Once you think the concept of shifting of the origin in two dimensions is clear to your
students, you could ask them in what way this could be extended to three dimensions.
You could give them a hint by doing activities with them of the kind suggested in the
previous section to make them comfortable with the 2D representation of 3D space.
Once you are assured that this aspect is taken care of, you could help the students to
extend this to the translation of axes in 3D. Here, you could again begin by asking a
student to come up on the board and pick any point as the origin, and draw the 3
coordinate axes through it. Then ask another student to pick another point 0' as the
new origin on the board (see Fig. 5).

K
Fig. 5

Suppose 0' has coordinates (a, b, c) in the coordinate system OXYZ and O'X', O'Y',
O'Z' are the axes through O'.parallel to OX, OY and OZ, respectively. As in the 2D
case, ask your students if they can prove the following relationship geometrically:

Why don't you try some exercises now?

ElO) How would you help your students to see the utility of a translation of the
axes in 3-space?
E l 1) What kinds of activities would you give your learners to assess their
understanding of the geometric aspect of translation?

Translation is not the only ,kind of-shift in axes that the students encounter. Let us
consider another rigid body transformation of the coordinate system.
10.3.2 Rotation
As in the case of translation, rotation would be easier for your students to understand
if you first do it in the 2D case. To start with, you could ask your students if they can
think of any other way of transforming the axes. Maybe, someone would come out
with the idea of rotating the axes. If so, you could ask herlthem to explain how this
Geometry and Mechanics
works to the rest of the students. Otherwise, you could prod them by asking each of
them to do the following task.

Activity: Take a redtangular sheet of paper and pick any point on it as the origin 0.
Draw the coordinate axes OX and OY through it. Now, draw OX' and OY' through 0
so that LXOX' = LYOY' = 0, say. (Here each child could pick the angle she wants.)
Now, she should chpck that X'OY' is a new coordinate system with coordinate axes
OX' and OY', and qrigin 0.

Next, pick any point P on the sheet of paper (see Fig. 6). From P, draw
perpendiculars PL on OX and PA on OX'. Also, draw from A a perpendicular on OX,
namely, AM. Then draw AB perpendicular to PL. Ask them which of these
segments PL, AM and AB are parallel to OX or OY. Can they find L BAO and
LPAB? Do they rbalise that LBAO = 0, L PAB = 7d2 - 0?

Fig. 6 '

Continue with the activity, asking them to find x and y in terms of x', y' and 8. They
could do this task in groups if they feel the need to. Once you have given them
enough time to di$cuss this, you could ask them to come up and present their so!utions
to the others on tHe board.

How many of them get the following solution?

x=OL=OM-AB=OAcos0-PAsin@=x'cos0-y'sin0
y = PL = BP + AM (here BL = AM)

= PA cos 0 + OA sin 8 = y' cos 8 + x' sin 8

What other solutions emerged?

Having obtained

x =xfcosO- yfsin8
y = x'sin 8 + y'cos 8

you could ask your students how they would find (x', y') in terms of (x,y). How many
of them solved equations (4) for x' and y' to obtain the values of these in terms of x,y
and 0? How many said, "We can get x and y by rotating the axes in the reverse
direction through the same angle. This means that we put -8 in (4), to get the required
equation."?

' How many o t h d routes did the students use to get the following equations?
 Introducing 3D-Geometry

As in the case of translation, your students need to get a geometric feel for what is
going QJI when these transformations are applied. How would you help them get this We have discussed
rigid body motions in
feel? How would they realise that rotation is also a rigid body motion. Through Sec. 12.2 in detail.
examples, you could let them see that a straight line will remain a straight line, a circle
will remain a circle and a conic section (parabola, ellipse, hyperbola) will remain the
same conic section under a rotation. Only its equation will change. And, again, as in
itanslation, you could choose examples to help them see that with a proper choice of
the coordinate system the equations could be put in standard form. Such forms can
then help in studying the objects represented by any given equation. For instance, ask
your students to tell you the object represented by the equation

I
11x2 + 2&XY + 9y2 = 12(x& + y + 1).
How many of them can do it? But, if you now tell them to rotate the axes through an
x2 y 2
angle of 30°, and then shift the origin to (1,0), they will get -+ -= 1. 'This would
2 3
be recognisable to them as an ellipse.

Here are some related exercises for you.

. .
E12) How would you help students to appreciate the underlying geometry taking
place when an equation is transformed under a rotation of the axes?

E13) If T is a translation and R a rotation, is TOR= ROT?Give reasons for your

answer.
1
O k e students are familiar with the rotation of the axes about the origin in two
dimensions, we can encourage them to extend this to three dimensions. Some of them
may do this in inany different ways. Some generalisations that students make may
really surprise you! However, if the students aren't able to extend the rotation to 3D
you could give them some 'concrete' help as follows.

You could ask your Students to take any cuboidal box, as shown in Fig.2, and rotate it
about the origin. Most students rotate the box in such a manner that OA and OB turn,
We have discussed
but not OC. This results in a rotation in the XOY plane, i.e., a two-dimensional 'direction cosines' in
rotation. A few students may exhibit a general rotation in which OC also turns. They the next section.
I could share this with the rest of the class. To generate the transformation equations in
this case, the students would require familiarity with direction cosines. It is important
that your students realise that this rotation of the axes is also a rigid body motion. For
instance, the three coterminous edges of the box, which are mutually perpendicular,
will remain mutually perpendicular after a rotation.

Now, let us assume that the students can 'see' what is going on geometrically, and are
comfortable with the diagrammatic representation. The next step is to ask them to see
if they can generalise (4) and (5). 'They need to see how the equations of the
coordinates change when the axes are rotated. You may need to hint to them that they
should consider the direction cosines of the new axes w.r.t. the old axes. You could
ask each of them to draw the OXYZ and OX'Y'Z' systems on a sheet of paper (see
Fig. 7).
Geometry and Mechanics

Fig. 7 :Rotating the axes in three-dimensional space

Ask them to call the direction cosines of OX', OY' and OZ', respectively, 11, m,, n, ;12,
m2, n2 ;13,mj, n3 with reference to the old axes. If they call the coordinates of a point
P, (x,y,z) w.r.t. the original axes and (x', y', z') w.r.t the new axes, ask them to draw
perpendiculars from P to OY and OY', respectively, say PM and PN. Then ON = y' is
also the projection of the line segment OP on OY' with direction cosines 12, m2, n2.
Your students would need to know why this implies

You could ask your students to get similar equations for x' and z', viz.,

X' = llx + mly + n l z , and

z'=l3x+rn3y+n3z.

If they manage this,you could ask them to obtain (x, y, z) in terms of (x', y', z'). Do
they need help to see that with reference to OX', OY' and OZ', the direction cosines of
OY are ml, m2 and m3? In fact, using the fact that m j + m i + m: = 1, and the
equations above, they should be able to obtain

Similarly, ask your students to obtain

x =l,xl+ 12y'+ 13zf,and

Maybe, your learners find 2difficult to remember all these equations. Writing them in
the form of a table, as below, helps.

This could be taken as a table for transformation of coordinates through a rotation of

the axes, without shifting the origin. To find z', for instance, in terms of x, y, z, we
 Introducing 3D-Geometry
make use of the elements in the third row, multiplying these with x,y, z, respectively,
and add them to get the required equation. Similarly, if we have to find x, say, in
terms of x', y', z'. we use the elements in the first column, multiply them with x', y', z',
respectively, and add them all to get the required equation.

It is important that the students understand the implications of what we have

discussed. One is suggested in the exercise below.

E14) In 2D, we say 'rotate the OXY system through an angle 0 to get OX'Y". In
3D, why can we not say that we rotate OXYZ through an angle 0 to get
OX'Y'Z'?

We will discuss direction cosines in some detail later. Let us now consider how to
help students visualise various aspects of lines in 3-space.

10.4 LINES -SKEW AND OTHERS

I asked my niece, who is in Class 8, if she could give me an example of two parallel
lines, two intersecting lines and two lines which are neither parallel nor intersecting. Lines in 3-space which are
The first two examples were no problem -she pulled out a piece of paper and drew neither parallel nor
intersecting, are called skew
them. But, for the third example, she tried drawing them in various ways without
lines.
succeeding. Finally, I suggested that she look around the room and see if she finds
any such pair of lines. She thought for a bit, looked around the room, thought some
more and finally came out with an answer. What do you think it was? One line was
the edge where two walls met, and the other was where a third wall and the floor met
(see AB and CD in Fig. 8).

U
Fig. 8

She was right. This is the kind of example we should use to introduce students to
skew lines. You could also ask them to give you several other examples of skew and
non-skew lines. With such exercises they are likely to be able to visualise skew lines.
Try some exercises now.

E15) Ask your learners to identify at least 3 planes which pass through a point P
and are parallel to a line L. How many such planes are there?
El 6) Ask your students how many lines can be drawn through a point P, which will
i) intersect a line L?
ii) be parallel to L?
iii) which will not intersect L?
Also ask them the reasons for their responses.
 ~ e o m ~and
t r Mechanics
~
Another useful activity for students in this regard is the following.

Activity : Ask the student to pick any point in space, say P, and draw a line through P,
say, L. Ask her haw many lines can we have which pass through P and intersect L at

paper and draw a line L on it. Then hold another paper parallel to this paper at a
certain distance from it (see Fig. 9). Next, ask her: How many lines on the second
paper will be parallel to L? How many lines on this sheet will be skew to L?
-,
Do your students tealise that only one line on the second sheet can be parallel to L and
infinitely many are skew? What is their reasoning behind this? What proof can they
Fig. 9 : L is parallel to give?
L,, and skew w.r.t LZ,
L.3,... Some students may reason that the lines can't intersect because they are on parallel
planes. But why are they not parallel then? Here is where they would need to find the
angle between any two lines in space. This requires them to use tHe concept of
direction cosines. To help them understand this concept, the following activity may
be useful.

Activity : Divide your students into groups of two. Ask each group to hold a pencil
up at an angle to the surface of their desks and at a corner of their desks. You could
ask them to take the 2 adjacent corners of the desk as the x and y axes and a
perpendicular to these corners as the z-axis (see Fig. 10). Then ask them to find the
angles that the pencil makes with each of the three axes, say, a , p, y .

Is it easy for them to find the angles? Do they try to hold a protractor up to do so?
What other strategies do they use? Once they have been given ample time to explore
the situation on their own, you could tell them how to find the angle between two
lines.

For this, it would be better to take them back to 2D. Ask them how they find the angle
between 2 lines in 2-space. It is in terms of the slopes of the two lines. Why is the
Fig:lO ' slope of the line used as the identifier? Let's see. If we assume that the origin is one
point on the line, pnd the other is P(xl,yl), then the slope is tan-' (yl/xl) . So,
knowing xl and y, allows us to pinpoint the line.

Now ask the studmts how this can be extended to 3D.' Given any line L in 3-space,
how do we identify it uniquely? Do they need a hint? If yes, ask them to assume that
L passes through she origin 0 , and P(xl,y,,zl)is another point on it. Then if
a , p , y are the angles L makes with the positive direction of the x, y and z axes,
respectively, ask them to find these angles in terms of X I ,yl and zl.
Do they get the fdHowing equations?

1 Y1
cos a = -,cosp=-,cosy=- z1
lop1 lop1 lop1
Do they now realise, how the direction cosines cos a , cos p and cosy uniquely
determine L? Can they also use the equations above to prove that
cos2a + cos2p + cos2y= I ?

Now, ask your stbdents if they can suggest a method for finding the angle between any
two lines in space. Do they think in terms of the direction cosines (dcs, in short)? If
so, in what way? You may need to tell them to pick any convenient point in space as
the origin, 0, ~ d r o u 0~ ,h they should draw lines parallel to the two given lines.
Then the angle between the two given lines is defined as4he angle between tJe .
lines parallel to them drawn through 0.

You could,ask your students questions like :

 Introducing 3D-Geometry
What happens to this angle if the given lines are parallel?
(Do they realise that in this case the lines drawn parallel to them through a
'
point will coincide, so that the angle between them will be zero?)
tx
If a line L makes an angle a with the x-axis (as shown in Fig. 1l), then why is
- cosa also a direction cosine of L?
(As L extends indefinitely in both directions, it makes an angle IT- a with the
x-axis. Therefore, cos(x- a ) is also a direction cosine of L. Similarly, if the
z
angles made with the positive directions of the y-axis and z-axis are P and y.
respectively, then IT-P and h-y are also angles made with the positive
directions of these axes. Therefore, if the direction cosines of the line L are
cos a , cos P, cos y, then so are - cos a , - cos P, - cos y.)
dc is short for 'direction
If a,b,c are any 3 numbers such that a2+b2+c2=1,can you find a line L with dcs cosine'.
a,b,c?
(Let P have coordinates (a,b,c). Then, can they show that (OPI
=1, and that
OP is the unit vector along the required line?)
The students could also be
a b c 2 2
If a,b,c are 3 numbers such that - = - = - , where 1 + m +n 2
= 1 , then told that denoting the
direction cosines of a line.
l m n
by I, rn, n is a convention.
a b c
prove that 1 = + n =f

Here the same sign, + ve or - ve, is to be taken with all of them. (Here is where you
could tell them what direction ratios (drs, in short) are.)

Some more exercises that you and your students could do are given below.

E17) Are the direction cosines of a line also direction ratios of this line? Give
reasons for your answer.

E18) Find the direction cosines of a line which is equally inclined to the three
coordinate axes.

E19) If a line makes angles a,P,y,G with the four diagonals of a cube, prove that

E20) Suppose 11,ml, nl ;12,m2, n2 ;13, m3, n3 are the direction cosines of three
mutually perpendicular lines. Show that 11, 12, 13; ml, m2, m3; nl,n2,n3 are also
the direction cosines of three mutually perpendicular lines, and

E21) Find the direction cosines of the vector i - 2 j + 3k.

Now, how would your students find the distance between any two lines in 3-space?
Given any two lines L1 and Lz, the distance between them is the shortest distance
among all the distances PQ, where P ranges over all points on L1 and Q ranges over all
points of L2. (Why?) So, if L1 and L2 intersect, the distance between them is zero, this
Geometry and Mechanics being the distance at the point of intersection. If L1 and L2 are parallel, the
perpendicular distance between them always remains the same. In fact, this distance
is the shortest distance between any two points that lie on the two different lines, as
shown in Fig. 12.

Now, what is the distance if L, and L2 are skew? Ask your students to think of ways
of finding this. Here you could hint to them that given any two skew lines, we can
find two planes which pass through the two lines respectively, and are parallel to each
other. How this can be done by transforming the axes suitably is given in the solution
below.

Problem 1: Giveu a pair of skew lines, reduce their equations to the simplest possible
form.

Solution : Let AB and CD be two skew lines and let LM be perpendicular to both the
lines, where L is on AB and M is on CD. Then LM is their line of shortest distance.
Fig. 12 : AB is the line We suppose that this shortest distance is 2c. Choose 0 as the middle point of the line
of shortest distance. segment LM. Through 0 we draw two lines OE and OF parallel to CD and AB, as
shown in Fig.13. If 28 is the angle between AB and CD, then L EOF = 28.
+-

We choose the bisectors of the angles between OE and OF as the x-axis and y-axis, as
shown in Fig.13, and ML as the z-axis.. Note that the three lines OX, OY and OZ are
% 71:
mutually perpendicular. The line OF makes angles 0, - - 0,- with the axes OX, OY
2 2
and OZ, respectively. Therefore, the direction cosines of AB, which is parallel to OF,
are cos 0, sin 0,O.
71:
Similarly, the line OE makes angles 0,:
L
+ 0,;71: with the x-axis, y-axis and z-axis,
L

respectively. Therefore, the direction cosines of the line CD are cos 8,- sin 0,O.
Also the coordinates of the points L and M are, respectively, (0, 0, c) and (O,O, -c).
Therefore, the equations of the lines AB and CD are

x-0 y-0
-- - -, z = -c, i.e., y = -x tan 0, z = -c, respectively.
C O S ~ sin 8

These are of the form y = mx, in the parallel planes z = c and z = -c.
 Introducing 3D-Geometry
Once students assume that the skew lines can lie in parallel planes, ask each student to
draw two skew lines on two sheets of paper (as in Fig. 9). Then she should use a
pencil or a long thin stick to find the different perpendiculars between the planes, and
the one giving the shortest distance between the two lines. In fact, from different
points of either one of these lines, we can draw perpendiculars to the parallel plane
containing the other line. One and only one of these perpendiculars will have its foot
at a point lying on the other line. But the lengths of all these perpendiculars will be the
same. The perpendicular which intersects the two lines is along the line of shortest
distance between the two skew lines. This will give the distance between the two
skew lines.

Try some related exercises now.

x-1 - y-2 - 2-3

E22) Find the shortest distance between the lines -- -- -and
2 3 4

Let us now discuss another difficulty many students face while studying geometry.

10.5 LOCUS

I Through our interaction with teachers and students of Classes 9 to 12, we realised that
the moment students have to solve a problem on 'locus', they become afraid. In this
section we discuss ways of removing this problem.

As always, it is better to introduce students to this concept in 2D first. To start with,

you could ask each student to mark a point on a blank sheet of paper. Then, she
should move her pencil in such a way that its distance from this point remains the
same. Does each student draw a circle? You could then bring the students' attention
to the fact that they have just drawn the path traced by a point subject to a given
1
condition. Here you could introduce the term 'locus', telling them that it is just
another word for 'the path traced by a point subject to some given conditions'. You
I
could also relate this path to the set of all points that satisfy the given conditions.
So, for example, the set of all points that are at a given distance from the given point is 'Locus' is a Latin
the circle they have drawn. word, meaning
'place'. The plural
of 'locus' is 'loci'.
As in the case of any concept, you could introduce your students to loci by giving
them several related exercises in two-dimensional space. You could ask them to find,
I for example, the locus of a point that is always equidistant from the x and y axes.
You also need to do 'reverse' exercises like drawing aqwo-dimensional curve in
standard form, say, a line, or an ellipse, and then asking them to formulate a 'locus
) problem' to which this would be the answer. Such examples would help them become
familiar with the idea in 2D.

Loci are not only of points, but also of lines, or segments. Students could be asked to
find the locus in 2D of a line parallel to a given line. Do they see that this set of lines
is actually the whole plane? Give them several exercises related to finding the loci of
lineslsegments. One is given below, which you should also try.

E23) a) Draw the locus of a line segment PQ passing through a fixed point 0.
b) Which misconceptions of your students became clear to you after looking at
their solutions of (a) above?
Geometry and Mechanics Once students are comfortable with the algebraic and geometric views of loci in 2-
space, you could Gxtend the concept to 3D. The important point here is to bring home
the understandingithat the locus of a point, subject to one or more constraints, would
be different in 2Dand in 3D. For instance, ask them to find the locus of a point which
moves such that it is at a constant distance from a fixed point, in 2D first, and then in
3D. In two dimensions, they would find that the locus is a circle. However, in three
dimensions the locus is a hollow sphere (see Fig. 14).

fig. 14 : Locus in 2D and in 3D ,under the same constraint

Your students need to understand this difference. How would you help them in this
matter? One way is to show them many points in 3-space which are not on the circle,
but are equidistant from the centre. For instance, they can see this by taking a ball that
is cut into two semi-circles.

You could, similady, ask the students to find the 'locus' in 2D, as well as in 3D, of a
point which is equidistant from two points (see Fig. 15).

Fig.15 :The locus of a point which moves such that it is equidistant from two fixed points A and B
is
a) the right bisector of AB in 2-space;
b) a plane which is perpendicular to, and bisects, the line segment AB in 3-space.

The immediate answer you may get is the middle-point 0 of the line segment AB.
You may encourage your learners to think of some more points that are equidistant
from A and B. The learners would certainly arrive at the line bisecting the line .
segment AB at right angles, i.e., the right bisector of AB.

So far as plane geometry is concerned, this is fine. But if they are studying three-
dimensional geomt?try, then they need to go beyond this. You could give them a hint
by taking some other points, say PI,...,P,, lying on the plane passing through the
middle point 0 of the line segment AB, such that AB is along the normal to this plane.
Ask them to see if PI,...,P,, are equidistant from A and B. You could do this by
 ',' Introducing 3D-Geometry
standing up a cardboard piece as the plane and taking some points on it. By actual
measurement, they would see that any point P lying on the plane will be such that
PA = PB. Ask your learners to think about why this is so.

I You could also change your problem of the locus a little, so that your students
consider loci in vectorial form. For example, you can have your students do problems
like the following ones.

~rol;lem2 : A point P moves such that the direction of OP is always the same, say,
along a unit .vector a. Find the locus of P in vector form and in Cartesian form.

Solution : We take a point with positiowvector rl. Then, the condition of the
problem is shown in Fig. 16. From this figure, we see that
rl = sa,where s is a scalar.
As the point PI moves subject Yo this condition, s will keep changing. So, the
equation of the locus is given by
r = sa, SE R.
Your students need to realise twq things while solving the problem above. Firstly,
they need to understand that the second equation is the generalised form of the first
one. This is obtained by changing rl (the position vector of a particular point P I )to r
(the position vector of a.generic point P) on the locus.

Secondly, the students must understand which geometric object is representdd by this
equation. Do they realise that this is the equation of a straight line passing through the
origin 0 with the direction of the line along the unit vector a? Do they also notice
that the points on one side of 0 will correspond to positive values of s, and on the
other side to negative values of s?

You could tell them why the equation is called the parametric equation of the
straight line. The scalar s is called a parameter (it changes as the point moves along
the line).

In the Cartesian form, r will be (x,y,z), and a can be taken as (I, m, n). (Why?)
Then the vector equation will become ,

(x, y, z) = s(l,m, n).

This is equivalent to the three equations

x = s1, y = sm, z = sn.

These equations could be rewritten as

.------
x - Y ,z
l m n
- s,
which is the usual symmetric fdrm of the equation of a straight line, passing through
the origin, with direction ratios I, m, n.

Problem 3 : Find the locus of a line which is at a distance p from the origin and is
perpendicular to the unit vector n, in vectorial and Cartesian forms.

Solution : Let us assume that p > 0, i.e., 0 does not lie on the line L. Take any point,
- p h d t h position vectorrl, on L. The shortest (distancefrom 0 to L is the
perpendicdlar distance. If we assume that 0 is the initial point of n, then the shortest
distance lies along n.This is also the projection of OPI along n, and is given by OPl.n.
so, 6.X = p.
Geometry and Mechadics To obtain the equation of the locus, we take the set of all such rl (see Fig. 17). So, we
change rl to the position vector r of a current point P on the locus. The required
equation of the locus, therefore. is
\
---

r.n = p, which is the hormal form of the equation of a plane.

The Cartesian equivalent of this equation is obtained by putting r = (x, y, z) and

O n = (I, rn,n), where I, m, n are the direction cosines of n. (Note that, since n is a unit
vector, its coordinates are its direction cosines.)

Then (x, y,z). (l,m,n)=p, i.e., Ix + m y + n z =p.

,
4 1 7 :The plane Il X
is the locus of L.
You need to ask your learners to solve several problems like these. You could do the
following ones yourself and with them.

E24) Find the vector equation of the locus of a point which is always equidistant
from two fixed points.

E25) Find the equations of the locus of a point which moves such that its
perpendicular distances from two intersecting planes, given by the equations
a , x + b , y + c , z + d , =Oand a 2 x f b , y + c 2 z + d 2 =(),areequal.
What is the corre&onding problem, and the equations formed, in two-
dimensional space?

E26) A variable plane is at a fixed distance p from the origin. It meets the
coordinate axes in A, B and C,'respectively. Through these points, planes are
drawn parallel to the coordinate'glanes. Find the locus of their point of
intersection. \

E27) Prove thaythe locus of the line which intersects the lines y - z =1, x = 0;
z - x = 1, y =O; x - y = 1, z = O i s x2 + y2 +z2-1 = 2(yz+ zx+xy).

With this we end our discussionon problems related to studying 3D-geometry. We

shall continue our study on the use of 3D in mechanics in the next two units. But first,
let us take a brief look at what we have done here.

SUMMARY
In thisunit, we have focussed on the following points.
1. How to help familiarise students with the conventions used for representing
three-dimensional objects in two dimensions. With enough activities of the
kind suggested, students would find it easier to visualise 3D in 2D.
2. The choice of the origin and the axes is arbitrary. Given a situation, they can be
so chosen that the equations involved are reduced to the simplest possible form.
3. The equation representing a rigid body is altered by a translation or rotation of
the axes..However, the shape and size of the rigid body is not altered under such
transformations.

4. How to help students visualise lines in 3D, and pairs of skew lines in particular.

5. The relationships involving direction cosines, in various cases.

 Introducing 3D-Geometry
6. How to help students become comfortable with problems requiring them to find
loci, in 2D or in 3D, in vectorial or in Cartesian forms.

t
10.7 COMMENTS ON EXERCISES
El) (i) The coordinates of the points A, B, C, A', B', C' and 0 are (x,O,O),
(O,y,O), (O,O,c), (O,y,z), (x,O,z), (x,y,z) and (0,0,0), respectively.
1

(ii) The feet of the perpendiculars to the xOy, yOz and zOx plane ha$e their
co-ordinates (-1,2,0), (0,2,4); and (-1,0,-4), respectively.

(iii) 5

(iv) "(-3, -2, 1)

E2) A good way is to ask them to divide into groups, and make cardboard cut-outs
of these planes. Then they could intermesh the three cut-outs to see how the
octants form.
The students could alp0 be asked to choose different points in the classroom as
origins and identify the co-ordinates of different points in the room w.r.t. the
new origin. You could,get them to focus on how, as the origin changes, the
coordinates of a point &.r.t. the origin also change, and they fall in different
octants.

E3) Over here ypp could do exercises with two objectives:

i) showing that the origin can be any point;

ii) familiarising students with negative coordinates.

For instance, ask a student to pick any point in the room as the origin. With
reference to this point, ask another student to give 3 coordinate axes. Then,
t ask dther students to give points corresponding to, say, (-1,0, O), (1, -2, l),
and so on.

I E4) In 23, one can think of examples like graphing the relationship between the
distance a plane flies, and the time it flies,,if its velocity is 300 krnhr: Then, a
convenient choice of unit length along the axes would be 1 and 300,
respectively.
I
Think of similar examples in 3D. ,

To introduce the students to the vectorial approach, you need to start by taking
unit vectors i, j, k along the axes. Then you need to show that P(x,y,z) is also
given as OP = xi + yj + zk, where x, y, z are the projections of OP along the
axes. You could do this by giving the students several exercises of drawing
vectors from 0 , finding their projections along the axes, and cornparir~gthese
with the Cartesian coordinates of the endpoints. Do they see the relationship?
It may help to do such exercises in 2D first, and then extend them to 3D.
The direction of a vector of magnitude 0 is indeterminate because its initial
and terminal points coincide. Thus, 0 can have any direction.
I
I E7) 0 is a scalar and 0 is a vector with magnitude 0. What students must realise is
why an equation like a = 0 is meaningless, one side being a vector and ,the
other a scalar.
1
E8) i) For shifting the origin to (1,-I), the transformation equatiqns pi%
I
,Geomeh$ awl Mechanics
Uhder this translation, the equation 2x-3y-5 = 0 becomes
2(xf + 1)- 3(y' - 1)- 5 = 0 , i.e., 2x' - 3y' = 0.
Ask the students : Does the translation mean that the line has also
shifted? Was the original equation a line too? etc.
ii) The transformation equations are

Under this transformation, the given equation becomes

E9) The gived equation can be rewritten as

(x+112 + ( y - 4 ) 2 - 1 = o

If you shikt the origin to (-1,4), this becomes xf2+ Y'2 = 1.

Ask your students what the object presented by the original equation was?
Has its shape or size altered under the translation?

El01 Think of situations in which the equations become simpler to deal with after a
translation of the axes. E.g., see Problem 1 (given after E21 of this unit).
I
E l 1) Here, you would need to see if they understand that though the equation
representing a rigid body changes under translation, the body itself does not
change size or shape. You could draw an object, and then ask them to show
what the ~ b j e cwould
t look like after a translation, for example.

E12) The point about rotation of axes is that while the odgin remains fixed, the axes
in 3 dimansions chadge. You need to help your students appreciate that the
objects in space change their position relative to the-axes, but not relativk to
the origin, nor wtfh reference to each other. As a risult this transformation of
axes does not alter anything in the body. One activity could be to get your
students to pl* things on their desks, taking the axes along the edges of the
desk. Then, without moving the objects, rotate the edges without moving the
axes, showing the new axes by strings or lines drawn on the desks. They
would see that everything remains the same relative to each other; only the
coordinates of the point, obtained from projections on the 3 axes, change.
E13) Take a,&@nslation and rotation i n ' 2 to
~ start with, and'see why this does not
hold in gleneral. Then you can take examples in 3D to see why this is not true.

E14) Is it possible to always have LXOX' = LYOY' = LZOZ'= O? Think about

the relatipnship between the Is, ms and ns. Also see E20.

1 5 , Can the students show you that there are an infinite number of planes passing
through 8 given point parallel to a given line? Can you show them why this is
>
so?
E16) i) , An infinite number of lines
ii) Only one line
iii) An infinite number of lines
How wobld you help them to come out B ith proofs of these facts?

Ell) The direbtion cosines are proportional to themselves. (What is the constant of
proporticbnality?) Therefore, they are also dra
If I, m, n are the dcs of the line, then I = m = n and 12+ m2+ n2= 1.

where the same sign, positive or negative, is to be taken throughout.

You could take the cube to be a unit cube with three coterminous edges as the
axes (see Fig.18). The four diagonals of the cube will have direction ratios
l,l,l;-l,l,l;l,-l,l;l,l,-1.

Fig. 18

Let I, m, n be the dcs of the line which makes angles a, fi, y and 6 with the
four diagonals. Then
I+,m+n -I+m+n I-m+n I+m-n
cos a = ,cosp= , COS y = ,toss=
A J5 fi
squaring and adding these, we get the result.

The cosines of the angles which the x-axis makes with the three lines are 11, 12,
13. Similarly the cosines of the angles which the y-axis and z-axis make with
the three lines are ml, m2, m3; nl, n2, n3, respectively.
With reference to the given lines the dcs of the axes are, therefore,
11, 12, 13; m,, m2, m3; nl, n2,ns.
Also, these are mutually perpendicular lines. The result follows.

The direction ratios of the given vector are 1, -2,3. So, its direction cosines

Let I, m,n be the dcs of the line of shortest distance. Since this line is
perpendicular to the given lines, we get

-1 2 -1
&
Eliminating I, m, n, we get 1 = - ,m=x,n=x.

The shortest distance is the projection of the join of any two points P and Q
(on each of the given lines) onto the line of shortest distance. Taking P(1,2,3)
1
and Q(2,4,5), we git the shortest distance = (2-1) 1 + (4-2)m+(5-3)n = -.
JZ
a) Many students may think that this is a disc with centre 0 and diameter
PQ. It is a disc. However, this is the set of all possible positions PQ can
Geometry and Mechanics take, with 0 lying on PQ. So, 0 can coincide with P, or with Q, or be in
betweem. All these cases need to be covered. So it would be a disc with
centre O and radius PQ.

b) For instance, did some students just draw one segment? And, if so, what
did this indicate to you of their understanding of 'locus'?

Let's say the fixed points are A and B. You can take the middle point of the
line segmnt AB as the origin of reference, with the points A and B having
position vectors -a and p. This choice makes the equation simpler. If the
position vector of P1 is rl,,then the equidistance condition of the problem
gives us

You could help your students deduce that this means a is orthogonal to rl.
All points that are equidistant from A and B have a position vector that is
orthogonal to a.The equation of the locus, therefore, is obtained by changing
rl to r, the position vector of a current point on the locus. Hence, the'
required equation of the locus, is r.a = 0.

In fact, this is the equation of a plane passing through the origin and normal t
the vector a.

If (XI,yl, zl) is a point on the locus, then

*--
The required equations, therefore, are obtained by generalising (xl, yl, zl) to
(x,y,z), i.e.,
alx+b y+clz+dl -+Ia2x+b2~+~2z+d2l
, which is a line.

In 2-space the problem becomes findkg the locus of a point which moves
such that its perpendicular distance from each of two lines alx + bly + cl = 0
and a2x + b2y + c2 = 0 are equal. This locus could be found on the same ljnes
as above.
A(a,O,O), B(O,b,O), C(O,O,c) lie on the plane.
So, the equation of the plane is -
X
+-
Y
+ -Z = 1.
a b c
The planes are x = a, y = b and z = c. Their point of intersection is (a,b,c).
We know that as the plane varies, its distance from (0,0,0) remains p. So,

Therefore, the locus of (a,b,c) is

 -

Introducing 3~k&metry
E27) The planes through .the giveelines are

(y-z- l)+ax=O,(z-x- l)+by=O,(x-y- l)+cz=O. (1)

These planes will intersect in a line if

Eliminating a,b,c from (1) and (2) gives us the locus, which is the given one.