Analytic solid geometry by shanti narayan (2)

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ANALYTICAL SOLID GEOMETRY

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ANALYTICAL

SOLID

GEOMETRY
FOR

B.A. and B.Sc. (PASS and HONS.)

BY

SHANTI NARAYAN,

M.A.,

Principal,

Hans Raj

College, DcIhi-6.

(Delhi University)

Mr. N. Sreekanth
M.Sc.lMaths) O.U.

TWELFTH EDITION
Price

:

Rs. 6-00

1961

S.

CHAND

& CO.

DELHI-JULLUNDUfe-LUCKNOW

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Differential Calculus

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Modern Pure Geometry
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A Text Book of Matrices
A Text Book of Vector Algebra
A Text Book of Vector Calculus
A Text Book of Cartesian Tensors
Theory of Functions of a Complex Variable
A Text Book of Modern Abstract Algebra
A Text Book of General Topology ( Under preparation)

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PREFACE TO THE TWELFTH EDITION

A

Chapter on General Equation of the second degree and reduccanonical forms and classification has been added. It is
the treatment is natural and simple and as such will

that
hoped
appeal to the imagination of the students.
tion to

Hans Raj

SHANTI NARAYAN

College,

Delhi University,

January, 1961.

PREFACE TO THE FIRST EDITION
This book is intended as an introduction to Analytical Solid
Geometry and covers as much of the subject as is generally expected
of students going up for the B.A., B.Sc., Pass and Honours examinations of our Universities.

have endeavoured to develop the subject in a systematic and
manner. To help the beginner, elementary parts of the
subject have been presented in as simple and lucid a manner as
possible and fairly large number of solved examples to illustrate
various types have been introduced. The books already existing
in the market cover a rather extensive ground and consequently
I

logical

comparatively lesser attention is paid to the introductory portion
is necessary for a beginner.
The book contains numerous exercises of varied types in a graded
form. Some of these have been selected from various examination
papers and standard works to whose publishers and authors I offer

than

my

best thanks.

extremely indebted to Professor Sita Ram Gupta, M.A.,
of the Government College, Lahore, who very kindly went
through the manuscript with great care and keen interest and
suggested a large number of extremely valuable improvements.
I

am

P.E.S.,

I shall be very grateful for any suggestions for
corrections of text or examples.

improvements or

LAHORE

SHANTI NARAYAN

:

June, 1939.

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CONTENTS

CHAPTER I
Co-ordinates
Articles

Page
...

1

...

2

1*3.

Introduction
Co-ordinates of a point in space
Further explanation about co-ordinates
Distance between two points
Division of the join of two points

1-4.

Tetrahedron

...

1*5.

Angle between two

1-6.

1*8.

Direction cosines of a line
Relation between direction cosines
Projection on a straight line

1'9.

Angle between two

1*1.

1-11.
1*2.

1*7.

lines
.

lines in terms of their direction cosines
Condition of perpendicularity of two lines

CHAPTER

1

...

...

3

...

4
6

.

...

7

...

7

...

7

...

9

...

12

...

19

...

19

...

20

...

22

...
...

23
24

....

27

...

2&

...

30

...

31

...

32

II

The Plane
2-1.
2-2.

2-3.

2-4.

2-42.
2-5.

26.
2- 7.

2'8.
2-9.
P

2 10.

Every equation of the first degree in x
Normal form of the equation of a plane

t

y, z represents

a plane

Transformation of the general equation of a plane to the normal
form. Direction cosines of normal to a plane
Determination of a plane under given conditions. Equation of a
plane in terms of its intercepts on the axes

Equation of tho plane through three given points
Systems of planes
Two sides of a plane
Length of the perpendicular from a given point to a given plane.
Bisectors of angles between two planes
Joint equation of two planes
Orthogonal projection on a pla.ne. Projection of a given plane
area on a given plane
Volume of a tetrahedron in terms of the co-ordinates of its
vertices

CHAPTER III
Right Line
3- 1.

3*2
3*3.

line.
Equations of a' straight line in terms of its
direction cosines and^the co-ordinates of a point on it. Equations of a straight line through two points. Symmetrical and
unsymmetrical forms of the equations of a line. Transforma*

tion of the equations of a line to the symmetrical form.

Equations of a

Angle between a line and a plane

The

condition that a given line

may

lie in

a given plane

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,

...

37
42

....

43

...


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Articles

Page

3'4.

The condition that two given

3*5.

Number

3*6.

lines

may

intersect

of arbitrary constants in the equations of a straight line.
Sets of conditions which determine a line
The shortest distance between two lines. The length and
equations of the line of shortest distance between two straight
lines

3' 7.

3'8.

Length of the perpendicular from a given point to a given
Intersection of three planes. Triangular Prism

CHAPTER

line

...

44

..."

49

...

63

...

59

...

60

IV

Interpretation of Equations

Loci
4*1.

Introduction

4*2.

The equation to a surface
The equations to a curve

4*3.

...

64

...

64

65

4*4.

Surfaces generated by straight lines.

intersecting three given lines

4*5.

Equations of two skow

lines in

Locus of a straight

line

a simplified form

...

65

...

69

...

72

...

74

CHAPTER V
Transformation of Co-ordinates

5*2.

Introduction. Formulae of transformation. The degree of an
equation remains unaltered by transformation of axes
Relations between the direction cosines of mutually perpendi-

5 '3.

Invariants

5*1.

cular lines

Revision Exercises I

^CHAPTER

...

75

...

80

...

85

VI

The Sphere
6*1.

Definition and equation of the sphere

6*2.

Equation of the sphere through four given points
Plane section of a sphere. Intersection of two spheres

6*3.

6*4.

Equations of a

65.
6*6.

Intersection of a sphere and a line.
Tangent plane. Plane of contact.

6* 7.

Some results concerning poles and polars.

Angle of intersection of two spheres. Condition for two spheres

6*8.
6*9.

circle.

Sphere through a given circle
Power of a point
Polar plane.

86

...

88

...

89

...

93

...

95

...

103

Pole of a plane.

to be orthogonal
Radical plane. Coaxal system of spheres
Simplified form of the equation of two spheres

VCHAPTER

...

...

105

...

107

...

110

VII

The Cone and Cylinder
7*1.

of a cone, vertex, guiding curve, generators.
Equation of the cone with a given vertex and guiding curve.
Enveloping cone of a sphere. Equations of cones with vertex at
origin are homogeneous

Definitions

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Page

Articles
7*2.

7 -3.

7*4.

7-5.

Condition that the general equation of the second degree should
represent a cone
Condition that a cone may have three mutually perpendicular
generators
Intersection of a line and a quadric cone. Tangent lines and
tangent plane at a point. Condition that a plane may touch
a cone. Reciprocal cones

Intersection of two cones with a common vertex

7*7.

Bight circular cone. Equation of the right circular cone with
a given vertex, axis and semi -vertical angle
Definition of a cylinder. Equation to the cylinder whose
generators intersect a given conic and are parallel to a given
line.
Enveloping cylinder of a sphere

7 '8.

The

7-6.

Equation
right circular cylinder.
cylinder with a given axis and radius
Revision Exercises II

...

115

...

119

...

121

...

127

...

128

...

130

of the right circular
...

132

...

136

...

139

...

140

...

141

APPENDIX
Homogeneous Cartesian Co-ordinates
Elements At Infinity
cartesian co-ordinates

A'l.

Homogeneous

A'2.

Elements at

A' 3.

Illustrations

A*4.

Sphere in

A*5.

Relationships of porp3ndicularity.

infinity

Homogeneous

co-ordinates

CHAPTER

...

142

...

143

...

144

...

145

...

148

VIII

The Conicoid
8*1.

8*2.

8-3.

The general equation of the second degree and the various
surfaces represented by it.
Shapes of some surfaces. Nature of Ellipsoid. Nature of hyperboloid of one sheet. Nature of hyperboloid of two sheets
Intersection of a line and a central conicoid. Tangent lines and

tangent plane at a point. Condition that a plane
central conicoid. Director sphere
8*34. Normals to a central conicoid
8-4.
8*5.

Plane of contact
Polar plane, conjugate points,
linos,

8-61.
8-62.
8-71.

conjugate

may

planes,

touch a

:

153

.,.

158

...

159

...

162

...

163

...

164

conjugate

polar lines.

Enveloping cone
Enveloping Cylinder
Locus of chords bisected at a given point

...

Plane section with a

given centre
8*8.

Locus of mid-point8 of a system of parallel chords
Conjugate diameters and diametral planes

89.

Paraboloids

...

174

8*91.

Nature of elliptic paraboloid
Nature of hyperbolic paraboloid

Intersection of a line and a paraboloid

...

174

...

174

8-72.

...

166

...

167

'

8-92.
8*93.

5-94. Properties of paraboloids

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175

...

176


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Articles

Number

of normals from a given point to a paraboloid
8'96. Conjugate diametral planes
8*95.

...

178

...

179*

...

180

...

180'

...

181

CHAPTER IX
Plane Sections of Conicoids
9-1.

Introduction

9*2.

Determination of the nature of plane section of central conicoids
Axes of central plane sections. Area of the section. Condition
for a rectangular hyperbola
Axes of non-central plane sections. Area of parallel plane

9 '21.
9*3.

sections

Circular sections of an ellipsoid
9'41. Any two circular sections of opposite systems
9'4.

lie

on a sphere

...

18ft

...

189

...

190

9-42.

Umbilics

...

192

9'6.

Nature of plane section of paraboloids

...

193

9 '61. Axes of plane sections of paraboloids. Condition for a rectangular hyperbola. Area of a plane section. Angle between
the asymptotes of a plane section.
Circular sections of paraboloids
9*6.
9-61.

Umbilics of paraboloids

...

193

...

196

...

196

...

198:

CHAPTER X
Generating lines of Conicoids
10*1.

10-2.

Generating lines of hyperboloid of one sheet
Equations of generating lines through points

of principal

elliptic section

10-3.

10*4.
10*5.

10*6.
10'7.

10' 8.

10*9.

...

202
20$
206

. . .

20&

...

Projections of generators on principal planes
Locus of the intersection of perpendicular generators

Central point, line of strict ion and parameter of distribution for
generators of hyperboloid
Systems of generating lines of hyperbolic paraboloid
Central point, line of striction and parameter of distribution for
generators of hyperbolic paraboloid.
General Consideration. Generators of any quadric. Quadric
determined by three generators
Quadrics with real and distinct pairs of generators

10*10. Lines intersecting three lines

Revision Exercises III

...

...

211

...

213

...

215
216
217

...

221

...

...

CHAPTER XI
General Equation of the Second Degree
Reduction to Canonical forms

Appendix
Index

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...

224

...

269'


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CHAPTER

I

CO-ORDINATES
In plane the position of a point is determined by
obtained with reference to two straight lines in the
plane generally at right angles. The position of a point in space is,
however, determined by three numbers x, y, z. We now proceed to
explain as to how this is done.
Introduction.

two numbers

x, y,

Co-ordinates of a point in space. Let X'OX, Z'OZ be two
lines.
Through 0, their point of intersection,

1-1.

perpendicular straight

Z

Y'

e

M

X
O

Y
Fig.

1

XOZ

Y'OY perpendicular to the
three mutually perpendicular straight lines

called

the origin, draw a line

so that

we have

known

as rectangular co-ordinate axes.

plane

X'OX, TOY, Z'OZ

XOZ

(The plane
containing
the lines X'OX and Z'OZ may be imagined as the plane of the paper
the line OY as pointing towards the reader and OY' behind the paper).
The positive directions of the axes are indicated by arrow heads.
These three axes, taken in pairs, determine three planes,
;

XOY, YOZ and ZOX

ZX

or briefly XY, YZ,
planes mutually at right angles,
rectangular co-ordinate planes.

known

as

Through any point, P, in space, draw three planes parallel to the
three co-ordinate planes (being also perpendicular to the corresponding
axes) to moot the a.xes in A B, C.

y

Let

QA=x, OB=y and 0(7=?,
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2

These three numbers,

determined by the point P, are

x, y, z,

called the co-ordinates of P.

Any one of these x, y, z, will be positive or negative according
as it is measured from O, along the corresponding axis, in the positive
or negative direction.
three numbers,

Conversely, given

whose co-ordinates are
(f)

x, y, z.

To do

x, y, z, we can find a point
this, we proceed as follows :

Measure OA, OB, 00, along OX, 07,

OZ

equal to

x, y, z

respectively.
(ii)

Draw through

ZX

planes YZ,

and

A, B,

C

planes

XY respectively.

parallel to

The point where these three planes

the co-ordinate

intersect is the required

point P.
Note. The three co-ordinate planes divide the whole space in eight compartments which are known as eight octants and since each of the co-ordinates
of a point may be positive or negative, there are 2 3 = 8) points whose co-ordinates have the same numerical values and which lie in the eight octants, one in
(

each.

Further explanation about co-ordinates.

1*11.

In

1*1

above,

we have learnt that in order to obtain the co-ordinates of a point P,
we have to draw three planes through P respectively parallel to the

three co-ordinate planes. The three planes through P and the three
co-ordinate planes determine a parallelepiped whose consideration
leads to three other useful constructions for determining the coordinates of P.

The

parallelopiped, in question, has six rectangular faces

PMAN, LCOB PNBL, MAOC PLCM, NBOA
;

;

(See Fig. 1).
(i) We have
x=OA = CM=LP = perpendicular

y=OB=ANMP

z=OC=AM=NP
Thus

from
perpendicular from
= perpendicular from

P on the YZ plane

P on the ZX plane
P on the XY plane.

;

;

the co-ordinates x, y ; z of any point P, are the perpendicular
and
from the three rectangular co-ordinate planes YZ,

distances of

ZX

P

XY respectively.
(ii)

the line

Since

OA*

9

PA

lies in

the plane

PMAN which

is

perpendicular to

therefore

PBOB and PC

Similarly

OC.

P

Thus

the co-ordinates x, y, z of any point
are also the distances
the origin
of the feet A, B,
of the perpendiculars from the point

C

from

to the co-ordinate

*
plane.

A

line

axes

X'X, Y'Y and Z'Z

perpendicular to a plane

is

respectively.

perpendicular to every }jne in the

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Downloaded from />DISTANCE BETWEEN TWO POINTS
What

Ex.

co-ordinate axes

We

(Hi)

are the perpendicular distances of a point (x t y, z) from the
2
[Ans. ^/(y +z 2 ),
?

<

have

AN^OB =y
Thus

3

(Fig. 2) if

;

we draw

N

.and
plane meeting it at
meeting OX at A, we have

PN J_Z7
NA OY
\\

Exercises
1.

In

fig.

co-ordinates of
2.
3.

4.

x

co-ordinates of A, B,

C

;

M N when the

L,

,

z}.

a,

(i)

P

yb,z

x-avis, 2/-rm".v

(vi) z

by

= c.

locus of a point for which

2/=0, z=0,
?9

y

(v)

W hat is the
(iv)

6.

down the

Show that for every point (x, y, z) on the ZX plane, 2/ = 0.
Show that for every point (x, ?/, z) on the F-axis, #=0, z=0.
What is the locus of a point for which
= 0,
(i) :r
(n) 2/=0,
(Hi) z=0.
(iv)

5.

P

write
are (#, y,

1,

c,

(ii)

2=0, a;=0,

(v)

z~c,x=a,

any point (x, y, z), and a, p, y ar e
and z-axis respectively ; show that

(Hi] ic=0, 2/=0.
(vi)

the anyles

yb.
OP makes with

x~a,

which

cos a==x/r, cos p=7//r, cos Y=2/r,
inhere

rOP.

Find the lengths of the edges of the rectangular parallelopiped formed
7.

by planes drawn through the points (1, 2, 3) and (4, 7, 6) parallel to the coordinate planes.

N/T2.

[Ans.

Distance between two points.

the points P(XI,

y l9

Zj)

and Q(x 2

,

?/ 2 ,

To find

3, 5, 3.

the distance between

z 2 ).

Through the points P, Q draw planes parallel to the co-ordinate
planes to form a rectangular parallelopiped whose one diagonal is PQ.

Fig. 3

Then

APCM NBLQ LCPB QMAN SPAN, LCMQ
,

;

9

;

are the three pairs of parallel faces of this
parallelopiped,

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Downloaded from />ANALYTICAL SOLID GEOMETRY

Now _ANQ

line

a

is

Also

AQ

PA.

Therefore

lies

rt.

Therefore,

angle.

QMAN which

in the plane

is

perpendicular to the

AQPA.
Hence

Now, PA is the distance between the planes drawn through the
points P and Q parallel to the FZ-planc and is, therefore, equal to

the difference between their # co-ordinates.

AN=y

Similarly

2

~- z

,

lB

2

2

Hence
Thus

y1

2

NQ=z
PQ 2 -(x -x )H(y ~y ) + (z ~z

and

2

2

1

1

2

1)

.

the distance between the points
(

x i>

2/i,

and

*i)

(x 2

,

y 29

z2 )

is

Distance from the origin. When P coincides with the origin
so that we obtain,

*v/ Cor.

0,

we have x1 =^yl =z 1 =0

Note. The reader should notice the similarity of the formula obtained
for the distance between two points with the corresponding formula in
plane co-ordinate geometry. Also refer 1*3.

above

Exercises
1.

Find the distance between the points

2.

Show

(4, 3,

G)

and

(2,

3),

1,

[Ans. 7.

that the points

(0, 7, 10),

(

1, 6,

6),

(4,

9, 6)

form an

right-angled triangle.

Show that the three points (2, 3, 5), (1, 2, 3), (7, 0,
3.
Show that the points (3, 2, 2), ( 1, 1, 3), (0, 5, 6),
4.
sphere whose centre is (1, 3, 4). Find also its radius.
5.
(a, 0, 0),

are collinear.

(2, 1, 2) lie

on a

[Ans. 3.
Find the co-ordinates of the point equidistant from the four points
[Ana. (a, i&, Jc).
(0, 6, 0), (0, 0, c) and (0, 0, 0).

>/r3. Division of the join of two points.
of the point dividing the line joining
P(XI> Vi*

in the ratio

R

Let

L,

1)

isosceles

m

:

*i)

and Q(x2 y^
,

To find

the co-ordinates

za ),

n.

(x, y, z)

be the point dividing

PQ

in the ratio

m

Draw PL, QM, RN perpendiculars to the XY- plane.
The lines PL, QM, EN clearly lie in one plane so that
M, N lie in a straight line which is the intersection of
9

:

n.

the points
this plane

with the .XY-plane.

The
plane.

line

Let

through

it

R

intersect

LM shall lie in the
QM at H and K respectively.

parallel to the line

PL and

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same


Downloaded from />DIVISION OF THE JOIN OP TWO POINTS
The

HPR and QRK

triangles

we have

are similar so that

^^^^^^MQ-WR^z^z
_
~

m+n

by drawing perpendiculars
planes, we obtain

Similarly,

the

to

XZ and YZ
J

m+n

m+n

M

R

The point

divides PQ internally or
n is
externally according as the ratio
positive or negative.
Thus the co-ordinates of the point which
divides the join of the points (xl5 y^ Zj) and
n are
(#a UK Za) in the ratio

m

m

L
Fig. 4

:

m+n

m+n

\

:

m+n

/

m

:

n

:

/

In case

middle point.

Cor. 1. Co-ordinates of the
middle point of PQ, we have
:

I

R

is

the

I

:

so that

Cor. 2. Co-ordinates of any point on the join of two points.
Putting k for m\n, we see that the co-ordinates of the point R which
divides PQ in the ratio k
1 are
:

'

To every value

%

'

'

l+k

l+k

~l+k~

\

R

on the line
of k there corresponds a point
on the line PQ corresponds some value of k,

R

and to every point

PQ
viz.

PR/RQ.
Thus we

see that the point

fkx 2 +x
lies

on the

line

PQ

Tcyt+Vi

1

l+k

(

9

'

l+k

...

kzt+_zi\
l+k )

" iW

may have and

whatever value k

conversely

any

given point on the line PQ is obtained by giving some suitable value
to k.
This idea is sometimes expressed by saying that (i) are the
general co-ordinates of any point of the line joining P(x ly y lt zj and
Exercises

Find the co-ordinates of the points which divide the

/I.
points

4, 3),

(2,

(4,

(t)

6) in

5,

(1

:

-4)

and

(w) (2

[An*,
2.

triangle.

A

(3, 2, 0),

AD

B

(6, 3, 2),

C (-9,

6,

the bisector of the angle

co-ordinates of D.

line

joining the

the ratios
:

(i)

-3)

BAC,

1).

(n) (-2, 2, -3).
(4, -7, 6)
are three points forming a
meets BG at D. Find the
;

i

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An

'

(?f, fl,

H)-


Downloaded from />ANALYTICAL SOLID GEOMETRY

6
Find

3.

the ratio in

which the line joining
(2, 4, 5), (3, 5,

\a divided

by the YZ-plane.
general co-ordinates of any point on the line joining the given points are

The

'

This point will
zero,

the points

-4)

lie

'"

'

1 + fc
\l + k
on the YZ plane,

/

1-ffc
if,

and only

if,

x co-ordinate

its

is

i.e.,

Hence the required ratio= 2
point of intersection is (0, 2, 23).
/ 4.
Find the ratio in which the

:

(-3,

4,

3.

Putting

&=

2/3 in

(t),

wo

see that the

XY -plane divides the join of

-8) and

-6,

(5,

4).

Also obtain the point of intersection of the line with the plane.
[Ana. 2 ; (7/3, -8/3, 0).
2, 3,
3, 3) C(
points X(0, 0, 0), B(2,
3), are collinear.
Find in what ratio each point divides the segment joining the other two.
[Ana.
CA/AB^l.
Show that the following sets of points are collinear
6.
5.

The three

ABIBC=, BC/CA^2,

:

(t)

7.

(2,5, -4), (1,4, -3), (4, 7, -6).

() (5, 4, 2), (6, 2, -1), (8, -2, -7).
Find the ratios in which the join of the points

divided by the locus of the equation
3*2--722/2-j- 12822^3.
8.

AB and CD intersect.

Show

[Ana. -2 1 ; 1
are the four points ;
:

-4(4, 8, 12), J5(2, 4, 6), C(3, 5, 4), Z>(5, 8, 5)

that the lines

(3, 2, 1), (1, 3, 2) is

-2.

show

:

is common to the lines which join
-6) to (2, -5, 10).
Show that the co-ordinates of any point on the plane determined by
10.
the three points (x lt y^ z), (x 2 j/ 2 z 2 ), and (# 3 7/ 3 z 3 ), may be expressed in the
form

9.

(6,

-7,

that the point

0) to (16,

-19, -4) and
t

,

~~

11.

r=l,

(1,

1, 2),

(0, 3,

,

,

,

^
*

.

'

'

l+m-\-n
l+m+n
l+m+n )
Show that the centroid of the triangle whoso vertices are

(x r ,

yr

,

zr )

;

2, 3, is

/s
v

Tetrahedron. Tetrahedron is a figure bounded by four
It has four vertices, each vertex arising as a point of interplanes.
section of three of the four planes. It has six edges each edge arising
4
C 2 =6).
as the line of intersection of two of the four planes.
(
1*4.

;

To construct a tetrahedron, we start with three points A, B, C,
and any point D, not lying on the plane determined by the points
A, B, G. Then the four faces of the
tetrahedron are the four triangles,

ABC, BCD,CAD,ABD-,

the four vertices are the points
A, B, C,

D

and the
Fig. 5

six edges are the lines

AB, CD BC,
/>;

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AD

;

CA, BD.


Downloaded from />DIRECTION COSINES OP A LINE

CD

The two edges AB,

7

joining separately the points, A,

B

and

D are called a pair of opposite edges. Similarly BC AD and CA,
BD are the two other pairs of opposite edges.
C,

9

Exercises
four lines drawn from the vertices of any tetrahedron to the
a point which is at three-fourths of the
centroids of the opposite faces meet
distance from each vertex to the opposite face.
Show that the throe lines joining the mid-points of opposite edges of a
2.

The

1.

m

tetrahedron moot in a point.
1-5.
Angle between two lines. The meaning of the angle
between two intersecting, i.e., coplanar lines, is already known to the

We now give the definition of the angle between two nonstudent.
coplanar lines, also sometimes called skew lines.
Def. The angle between two non-coplanar, i.e., non-intersecting
lines is the angle between two intersecting lines drawn from any point

parallel to each of the given lines.
Note 1. To justify the definition of angle between two non-coplanar lines,
as given above, it is necessary to show that this angle is independent of the
position of the point through which the parallel lines are drawn, but here wo

simply assume this result.
Note 2. The angles between a given line and the co-ordinate axes are
the angles which the line drawn through the origin parallel to the given lino
makes with the axes.

^ 1'6.

Direction cosines of a line. If a, p, J be the angles which
makes with the positive directions of the axes, then cos a,
cos y are called the direction cosines of the given line and are

line

any

cos (3,
generally denoted

What

Ex.
/ 1*61.

A

by

I,

m, n respectively.

are the direction cosines of the axes of co-ordinates ?
[Ans. 1, 0, 0; 0, 1,0;

useful

relation.

be the origin

//

and

(x y

y

y

0, 0, 1.

z) the co-

ordinates of a point P, then

xlr,
where

m, n are

Z,

Through

OL=x.

that

P

ymr, z=nr

the direction cosines of

draw PLJ_#-axis

From

the

rt.

OP

y

and

r, is the

length of

OP.

so

angled

OLP,

triangle

we have

X
i.e.,

Similarly

.

=6

or

i

x=lr.

we have
2/=mr, z=nr.

^1*7. Relation between direction
direction cosines of any line, then
i.e.,

the

sum

Fig. 6

cosines.

// l,m and n are the

of the squares of the direction cosines of every line is one.

/>

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Downloaded from />ANALYTICAL SOLID GEOMETRY

8

OP

Let
that

Z,

07,

OZ

m

t

be drawn through the origin parallel to the given line so
are the cosines of the angles which OP makes with OX

n

,

respectively. (Refer Fig. 6)

Let

(x, y, z)

be the co-ordinates of any point

P on

this line.

Let

OP=r.
x=lr, y=mr, znr.
Squaring and adding, we obtain

But
!

Cor. If
cosines

Z,

ra,

a, 6, c

+m +n

2

2

=l.

be three numbers proportional to the actual direction

n of a

=

2

V=T T

we have

line,

=

==4 ~~~
2
a
8
V(0 +& +c )
XTf

where the same

sign, positive or negative,

is

to be chosen throughout.

From above, we

see that a set of three numbers
Direction Ratios.
which are proportional to the actual direction cosines are sufficient to
a line. Such numbers are called the direction
specify the direction of
Thus if a, 6, c be the direction ratios of a line, its direction
ratios.
*+

cosines are
Note. It is easy to see that if a line
Y, OZ, then the line
angles a, P, Y with OX,

OP ^through the
OP obtained by

origin

makes

producing

OP

Fig. 7

backwards through

Thus

make

will

angles

TT

a,

TT

3,

TC

Y with OX, OY, OZ.

if

cos a=Z, cos (3=w, cos
are the direction cosines of OP, then
COS(TC

a)=.

Z,

are the direction cosines of OP',

COS(TT
i.e.,

3)

=

the line

m,

Y=W
COS(TC

Y)

OP produced

/>
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n
backwards.


Downloaded from />PROJECTION OF A POlNf ON A
line, we can think of the direction
n determining the direction of one and the same
m,
This explains the ambiguity in sign obtained above.
line.
Note. The student should always make a distinction between direction
cosines and direction ratios. It is only when /, ?//, n are direction cosines, that
we have the relation

Thus

cosines

if

we

ignore the two senses of a

m, n or

/,

/,

Exercises
1.

6, 2,

are proportional to the direction cosines of a line. What are
?
[Ana.
(6/7, 2/7, 3/7).

,3

their actual values

What

2.

How many

The

3.

are the direction cosines of lines equally inclined to the axes?
?
[Ana. (l/\/3,
l/<\/3, dbl/V 3 ) J 4

such lines are there

-

co-ordinates of a point

P

are

Find the direction cosines

(3, 12, 4).

of the line OP.

The

4.

[Ana.

direction cosines

m,

I,

(3/13, 12/13, 4/13).

n, of two lines are connected by the relations
...($)

Q.

...(ii)

Find them ?
Eliminating n between

(i)

and

(ii)

we

;

get

or

This equation gives two values of l/m and hence there are two lines. The
two roots of (m) are 1 and j.
If Ii, m>i, ni and 1%, W 2 n 2 ^ e ^ ne direction cosines of two lines, we have

mi

2

in 2

Alsov
and

J^
2

.

+
'

2
'

+ 2* =0

or

.

The

(i)

find

them

two

linos are

/-5m+3n=0,

7/ 2 -f

_

~V^

ni

V6'

determined by the relations

Sw^-Sn^O

;

112

?

rx

,.v
(*)

[Ana.
,

1

1*8.

1*81.

cular

= 0,

mi

""Ve'

direction cosines of

.

1

__ _

"T -2 V6'

5.

+ +

.

x

'

-TT-T,

- 2 -,

\/14

\/14

1

3

-7T7

7^'

v!4

-yb

~7^
-yo

'

"7^
-yO

1

3

4

1

2

3

V26

V26

V26

V 14

A/1 4

V^ 4

Projection on a Straight line.
Projection of a point on a line. The foot of the perpendion a given straight line BC is called the
a given point

P from

A

/>
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