Analytical Solid Geometry - Shanti Narayan

THE

BOOK

WAS

DRENCHED

00

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

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PREFACE

TO

THE TWELFTH

EDITION

A

_Chapter

on

General

_Equation

_{of the second degree}

and

reduc-tion to canonical

forms

and

classification has

been

added. It is

hoped

that the

treatment

is natural

and

_simple

and

as such will

appeal to the imagination ofthe students.

Hans

_Raj

College,

SHANTI

NARAYAN

Delhi University,

January,

1961.

PREFACE

TO

THE

FIRST

EDITION

This

book

is intended as

an

introduction to _AnalyticalSolid

Geometry

and

covers as

much

of thesubject asis _{generally expected}

of students going

up

forthe B.A., B.Sc.,

Pass

and Honours

exami-nations of ourUniversities.

I

have endeavoured

to _{develop the} subjectin a _systematic

and

logical

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

toillustrate

various types

have

been

introduced.

The

books

_{already existing}

in the

market

cover a rather extensive

_ground

and

_consequently

comparatively lesserattentionis _paid to the _{introductory portion}

than

is _necessaryfor a _beginner.

The book

contains

numerous

exercises _{of varied types}ina

_graded

form.

Some

of these

have been

selected

from

various

examination

papers

and

standard

works

to

whose

publishers

and

authors I offer

my

best thanks.

I

am

_extremely

indebted to ProfessorSita

Ram

_Gupta,

_M.A.,

P.E.S., of the

Government

College, Lahore,

who

very kindly

went

through

the

_manuscript

with

_great care

and

keen

interest

and

suggested alarge

number

of

extremely

valuable

improvements.

I shall

be

_{very grateful for}

_any

_suggestionsfor

_improvements

or

correctionsoftext _{or examples.}

LAHORE

:

SHANTI

NARAYAN

CONTENTS

CHAPTER

I

Co-ordinates

Articles _Page

Introduction ... 1

1*1. Co-ordinatesofa_point in _space ... 1

1-11. _{Further explanation about}co-ordinates ... 2

1*2. Distancebetween twopoints ... 3

1*3. Division of the join oftwo points ... 4

1-4. Tetrahedron ... . 6

1*5. _Anglebetween two lines ... 7

1-6. Direction cosines ofa line . _{... 7}

1*7. Relationbetweendirection cosines ... 7

1*8. Projectiononastraightline ... 9

1'9. Angle between two lines in terms of their direction cosines

Condition_{of perpendicularity of}two lines ... 12

CHAPTER

II

The Plane

2-1. Everyequationofthefirst _degreeinxty,zrepresentsa plane ... 19

2-2. Normal formofthe equationofa_plane ... 19

2-3. Transformationof_{the generalequation}ofaplanetothenormal

form. Direction cosines ofnormalto_{a plane} ... 20

2-4. Determinationof_{a plane under given}conditions. _Equationofa

planeintermsofitsinterceptson the axes ... 22

2-42. Equationoftho _{plane through}threegivenpoints ... 23

2-5. _{Systemsof planes} ... 24

26.

Two

sidesofa plane .... 27

2-7. _Lengthofthe_{perpendicular}from_{a given point} to_{a givenplane.}

Bisectors of anglesbetweentwoplanes ... 2&

2'8. _{Joint equation}oftwo_planes ... 30

2-9. Orthogonalprojectiononapla.ne. Projection of a given plane

areaona given plane ... 31

2P

10.

Volume

of a tetrahedron in terms of the co-ordinates ofits

vertices ... 32

CHAPTER

III

Right Line

3-1. Equationsofaline. _Equationsofa'_straightline interms ofits

direction cosinesand^the co-ordinates ofa_pointon it.

Equa-tions ofastraightlinethrough two points. Symmetricaland

unsymmetrical formsofthe equationsofa line. Transforma* ,

tionof_{the equations}ofalinetothe_{symmetrical form.} ... 37

3*2 _Anglebetweenalineanda plane ... 42

Articles Page

3'4. Thecondition thattwo_givenlines

may

intersect ... 44

3*5.

Number

of arbitrary constantsin_{the equations of}a_straightline.

Sets of conditionswhich determine aline ..." 49

3*6. The shortest distance between two lines. The _length and

equations of thelineofshortest distancebetweentwostraight

lines ... 63

3' 7. _Lengthofthe perpendicularfrom agiven pointto_{a given}line ... 59

3'8. Intersection of threeplanes. Triangular Prism ... 60

CHAPTER

IV

Interpretation of Equations Loci

4*1. Introduction ... 64

4*2. Theequationtoasurface ... 64

4*3. The_{equations to a}curve 65

4*4. Surfaces generated by straightlines. Locus ofa_straightline

intersectingthree_givenlines ... 65

4*5. _Equationsoftwoskow linesina simplifiedform ... 69

CHAPTER V

Transformation of Co-ordinates

5*1. Introduction. Formulaeof transformation. The _degree of an

equationremainsunalteredbytransformation of axes ... 72

5*2. Relationsbetweenthe direction cosines of _mutually

perpendi-cularlines ... 74

5'3. Invariants ... 75

Revision Exercises I ... 80

^CHAPTER

VI

The Sphere

6*1. Definitionand _{equation of the sphere} ... 85

6*2. Equationof the_{spherethrough}four_givenpoints ... 86

6*3. Planesectionof a_sphere. Intersection oftwospheres ... 88

6*4. Equationsofacircle. Sphere through agivencircle ... 89

65. Intersection of a sphereandaline. Powerofa point ... 93

6*6. _Tangentplane. Planeof contact. Polar plane. Pole of aplane.

Some

results_{concerning poles}andpolars. ... 95 6* 7. Angleof intersection oftwospheres. Conditionfortwo_spheres

tobe_orthogonal ... 103

6*8. Radicalplane. Coaxal systemof spheres ... 105

6*9. Simplifiedformofthe equation oftwospheres ... 107

VCHAPTER

VII

The Coneand Cylinder

7*1. Definitions of a cone, vertex, guiding curve, generators.

Equation of the cone with agiven vertexandguiding curve.

Enveloping coneofasphere. Equationsof coneswithvertex at

Articles Page

7*2. Condition that the general equation oftheseconddegreeshould

representa cone ... 115

7-3. Conditionthatacone

_may

have three mutually perpendicular

generators ... 119

7*4. Intersection ofalineanda _quadric cone. _Tangent lines and

tangent plane at apoint. Conditionthataplane

may

touch

acone. _Reciprocalcones ... 121

7-5. Intersection oftwocones with a

common

vertex ... 127

7-6. Bightcircularcone. Equationof theright circular cone with

a_{given vertex, axis}andsemi-verticalangle ... 128

7*7. Definition of a cylinder. Equation to the _cylinder whose

generators intersect a given conicandare_paralleltoa_given

line. Envelopingcylinderofa sphere ... 130

7'8. The right circular cylinder. Equation of the right circular

cylinderwithagivenaxis

and

radius ... 132

Revision ExercisesII ... 136

APPENDIX

Homogeneous Cartesian Co-ordinates

Elements At Infinity

A'l. Homogeneouscartesian co-ordinates ... 139

A'2. Elementsat_infinity ... 140

A'3. Illustrations ... 141

A*4. Spherein

Homogeneous

co-ordinates ... 142

A*5. Relationships of porp3ndicularity. ... 143

CHAPTER

VIII

The Conicoid

8*1. Thegeneral equation of the second degree and the various

surfaces representedbyit. ... 144

8*2. _Shapesofsomesurfaces. NatureofEllipsoid. Natureof

hyper-boloid ofonesheet. Nature ofhyperboloidoftwosheets ... 145

8-3. Intersection ofalineanda centralconicoid. _Tangentlinesand

tangent planeata_point. Conditionthat_{a plane}

_may

touch a

central conicoid. Directorsphere ... 148

8*34. Normalstoacentral conicoid ... 153

8-4. Planeofcontact .,. 158

8*5. Polar plane, conjugate points, conjugate planes, conjugate

linos,polarlines. ... 159

8-61. Enveloping cone ... 162

8-62. Enveloping Cylinder ... 163

8-71. Locusofchordsbisected at_{a givenpoint}:Plane sectionwith a

givencentre ... 164

8-72. Locusof_mid-point8ofa_system of_parallel chords ... 166

8*8. _{Conjugate diameters}and _{diametral planes} ... 167

89. Paraboloids ' ... 174

8*91. Natureofellipticparaboloid ... 174

8-92. Natureof hyperbolicparaboloid ... 174

8*93. Intersectionofalineanda paraboloid ... 175

8*95.

Number

ofnormalsfroma_{given point to}a_paraboloid ... 178

8'96. Conjugatediametral planes ... 179*

CHAPTER

IX

Plane Sectionsof Conicoids

9-1. Introduction ... 180

9*2. Determinationof thenatureof_planesection of central conicoids ... 180'

9'21. Axesof central planesections. Areaofthe section. Condition

fora_{rectangularhyperbola} ... 181

9*3. Axes of non-central plane sections. Area of _{parallel plane}

sections ... 18ft

9'4. Circular sections ofanellipsoid ... 189

9'41.

_Any

twocircularsectionsof oppositesystemslieon a_sphere ... 190

9-42. Umbilics ... 192

9'6. Natureofplanesectionof_paraboloids ... 193

9'61. Axes of _plane sections of paraboloids. Condition fora

rect-angularhyperbola. Areaof a plane section. Angle between

theasymptotesofa_planesection. ... 193

9*6. Circular sections_{of paraboloids} ... 196

9-61. Umbilicsof paraboloids ... 196

CHAPTER

X

Generatinglinesof Conicoids

10*1. Generatinglinesof_hyperboloidofonesheet ... 198:

10-2. Equations of generating lines _{through points} of principal

ellipticsection ... 202

10-3. Projections of generatorsonprincipal planes ... 20$

10*4. Locusofthe intersection ofperpendiculargenerators ... 206 10*5. Centralpoint, lineofstriction_{and parameter}of distribution for

generatorsof_hyperboloid . . . 20&

10*6. _{Systemsof generating} linesof_{hyperbolicparaboloid} ... 211

10'7. Centralpoint, lineofstrictionand parameterof distributionfor

generators of hyperbolic paraboloid. ... 213

10' 8. General Consideration. Generators of _any quadric. Quadric

determined_bythree generators ... 215

10*9. Quadrics withrealanddistinctpairs ofgenerators ... 216

10*10. Linesintersecting threelines ... 217

Revision ExercisesIII ... 221

CHAPTER

XI

General Equation oftheSecond Degree

ReductiontoCanonical forms

Appendix ... 224

CHAPTER

I

CO-ORDINATES

Introduction.

In

_plane the _{position of}

a

_pointis

determined

_by

two

numbers

_{x, y,} obtained with reference to

two

straightlinesin the

plane generally at right angles.

The

position ofa pointin spaceis,

however,

determined

_by

three

numbers

x, y, z.

We

now

_proceed to

explainas to

how

thisis done.

1-1. Co-ordinates of a _point inspace.

Let

X'OX,

Z'OZ

be

two

perpendicularstraightlines.

_Through

_{0, their point} of intersection,

X

Y

Z

e

Y'

M

O

Fig. 1

called the _origin,

draw

aline

Y'OY

_{perpendicular} tothe

XOZ

_plane

sothat

we

have

three

_mutually

_{perpendicular straight}lines

X'OX,

TOY,

Z'OZ

known

_{as rectangular co-ordinate}axes. (The plane

XOZ

containing

thelines

X'OX

and

Z'OZ

_may

be

_imagined

as _{the plane of the}

_paper

;

the line

OY

as _pointing

towards

thereader

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

or _briefly

_XY,

_YZ,

ZX

_planes

_mutually

_{at right angles,}

known

as

rectangular co-ordinate planes.

Through any

point, P, in space,

draw

threeplanes parallelto the

three co-ordinate_{planes(being also perpendicular}to _{the corresponding}

axes) to

moot

the a.xes in

A

yB, C.

These

three

numbers,

x, y, z, determined

by

the point P, are

called the co-ordinates ofP.

Any

one

of these _{x, y,} z, willbe positive or negative according

asit is

measured from

O, along the corresponding axis, in the _positive

or_negative direction.

Conversely, given three

_numbers,

x,y, z,

we

can find a point

whose

co-ordinates are_{x, y,} z.

To

do

this,

we

proceed as follows :

(f)

Measure

OA,

OB,

00,

along

OX,

07,

OZ

equal to x, y, z

respectively.

(ii)

Draw

through

A, B,

C

planes parallel to the co-ordinate

planes

YZ,

ZX

and

XY

respectively.

The

_point

where

these three _planes intersect is the _required

point P.

Note.

The

three co-ordinate planes divide thewhole spacein eight

com-partments which are

known

aseight octantsand since each of the co-ordinates

of a _point

_may

bepositive or negative, there are23

(

=

8)pointswhose

co-ordi-nateshavethesamenumericalvaluesandwhichlieinthe eight octants, one in

each.

1*11. Further _explanation about co-ordinates.

In

1*1 _above,

we

have

learntthatin orderto 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

co-ordinates of P.

The

parallelopiped, in question, has sixrectangular faces

PMAN,

LCOB

;

PNBL,

MAOC

;

PLCM,

NBOA

(SeeFig. 1).

(i)

We

have

x=OA =

CM=LP

=

_{perpendicular}

from

P

on

the

YZ

_plane ;

y=OB=ANMP

perpendicular

from

P

on

the

ZX

plane ;

z=OC=AM=NP

=

_{perpendicular}

from

P

on

the

XY

_plane.

Thus

the co-ordinates _x,

_y

;z of

any

point P, are theperpendicular

distances of

P

from

the three_{rectangular co-ordinateplanes}

_YZ,

ZX

and

XY

respectively.

(ii) Since

PA

lies inthe plane

PMAN

which

isperpendicularto

theline

OA*

9 therefore

Similarly

PBOB

and

PC

OC.

Thus

the co-ordinates x, y, zof

any

point

P

arealso thedistances

from

the_{origin ofthe feet A, B,}

C

_ofthe_{perpendiculars}

_from

the_point

tothe co-ordinateaxes

X'X, Y'Y

and

Z'Z

_{respectively.}

*

_A

line _{perpendicular}to a plane is _{perpendicular to every }jne}in the

DISTANCE

BETWEEN

TWO

POINTS 3

Ex.

What

are the _{perpendicular} distances of a_{point (x}ty,z)fromthe

co-ordinateaxes? _{[Ans. ^/(y}2

+z

2

), </(z*+x*)t

<

(Hi)

We

have

AN^OB

=y

;

Thus

_{(Fig. 2) if}

we

draw

PN

_{J_Z7}

plane meeting it at

N

.and

NA

\\

OY

meeting

OX

at

_A,

we

have

Exercises

1. In fig. 1, write

down

theco-ordinates ofA, B,

C

;L,

M

,

N

when

the

co-ordinates of

P

are(#,y,z}.

2.

Show

thatfor_{everypoint(x, y, z)} on the

ZX

plane, 2/

=

0.

3.

Show

thatfor_{everypoint(x,}?/, z) onthe F-axis,#=0, z=0.

4. Whatisthe locusofapoint forwhich

(i) :r

=

0, (n) 2/=0, (Hi) z=0.

(iv) x a, (v) y by (vi) z

=

c. 5.

W

hatis_{the locus ofa point for}which

(i) 2/=0,z=0, (ii) 2=0,a;=0, (Hi] ic=0, 2/=0.

(iv)

yb,z

c, (v)

z~c,x=a,

(vi) x~a,

yb.

6.

P

?9 _{any point (x, y,}z),anda, p,y are theanyleswhich

OP

makeswith

x-avis,2/-rm".vandz-axis respectively; showthat

cos a==x/r, cos p=7//r, cosY=2/r,

inhere

rOP.

7. Find the lengths oftheedges of the rectangular parallelopipedformed

byplanesdrawnthrough thepoints (1,2, 3) and (4, 7, 6) parallel to the

co-ordinate planes. [Ans. 3, 5, 3.

N/T2.

Distance between

two

points.

To

findthe distance between

the_{points P(XI,}

_y

l9Zj)

and Q(x

2, ?/2, z2 ).

Through

the _{points P,}

_Q

draw

_{planes parallel to} the co-ordinate

planes to

form

a rectangular parallelopiped

whose

one diagonalis

_PQ.

Then

Fig. 3

APCM

,

NBLQ

;

LCPB

9

QMAN

;

SPAN,

LCMQ

are the three

Now

_{_ANQ}

is art. _angle. Therefore,

Also

_AQ

lies in_{the plane}

QMAN

which

is _{perpendicular}tothe

line

PA.

Therefore

AQPA.

Hence

Now,

PA

isthe distance

between

the planes

drawn

_through

the

points

P

and

Q

parallel tothe

FZ-planc

and

is, therefore, equal to

the difference

between

their

#

co-ordinates.

Similarly

AN=y

2

y

1,

and

_NQ=z

2~- zlB

Hence

_PQ

2

-(x

2

-x

1

)H(y

2

~y

1) 2

+

(z2

~z

1 ) 2 .

Thus

the distancebetween the_points

(xi> 2/i, *i)

and

(x2, y29 z2)

is

*v/Cor. Distance

from

the _origin.

When P

coincides with the_origin

0,

we

have x

1=^yl

=z

1

=0

sothat

we

obtain,

Note.

The

reader should notice the _{similarity of}theformula obtained

aboveforthe distancebetweentwopoints with the corresponding formula in

planeco-ordinategeometry. Alsorefer 1*3.

Exercises

1. Find thedistancebetween_{the points}(4, 3, G) and

(2,

1, 3),

[Ans. 7.

2.

Show

thatthepoints (0, 7, 10), ( 1,6, 6),

(4,

9, 6) form an isosceles

right-angledtriangle.

3.

Show

thatthethree points

(2,

3, 5),(1,2,3), (7, 0, 1) arecollinear.

4.

Show

that the points (3, 2, 2), ( 1, 1, 3), (0, 5, 6), (2, 1,2) lieon a

spherewhosecentreis (1,3, 4). Findalso itsradius. [Ans. 3.

5. Findthe co-ordinates of the _{point equidistant} from the four points

(a, 0, 0), (0, 6, 0), (0, 0,c)and(0,0, 0). [Ana. (a, i&, Jc).

>/r3.

Divisionofthe_joinof

two

_points.

To

_find the co-ordinates

ofthepoint dividing the linejoining

P(XI>Vi* *i)

and

Q(x

2,

y^

za ),

inthe ratio

m

: n.

Let

R

(x, y, z)

be

the point dividing

PQ

in theratio

m

: n.

Draw

PL,

QM,

RN

perpendiculars to the

XY-

plane.

The

lines

_PL,

_QM,

EN

_clearlylie in

one

_planeso thatthe _points

L,

M,

N

9liein

a

straightline

which

is theintersection of this _plane

with the .XY-plane.

The

line

_through

R

paralleltothe line

LM

shall liein the

same

DIVISION OF

THE

_JOIN

OP

TWO

POINTS

The

triangles

HPR

and

QRK

are similar sothat

we

have

^^^^^^MQ-WR^z^z

_

~

m+n

Similarly,

by

drawing

perpendiculars to

the

XZ

and

YZ

planes,

we

obtain

J

m+n

m+n

The

_point

R

divides

_PQ

_internally or

externally according as the ratio

m

:

n

is

positive or negative.

Thus

the co-ordinates of the_point which

dividesthe_{join of} the _points (xl5

y^

Zj)

and

(#a

UK

Za) intheratio

m

:

n

are

M

L

Fig. 4

\

m+n

m+n

m+n

/

/ _{Cor. 1. Co-ordinates of the middle}

point. In case

R

is the

middle _pointof

_PQ,

we

have

m

:

n

: : I : I

sothat

Cor. 2. Co-ordinates of _{any point} on the join of

two

points.

Putting k form\n,

we

see that the co-ordinates _{of the point}

R

which

divides

_PQ

inthe ratio k : 1 are

l+k

'

~l+k~

' '

l+k

\

%

To

_{every value} ofk there _{corresponds a point}

R

on

the line

_PQ

and

to _everypoint

R

on

theline

_PQ

_corresponds

some

valueofk, viz.

PR/RQ.

Thus

we

see thatthe _point

fkx

2

+x

1 Tcyt+Vi _{kzt+_zi\} ...

(

l+k

9

l+k

'

l+k

)

"

_iW

lies

on

theline

_PQ

whatever

value k

may

have

and

conversely

any

given point

on

theline

PQ

isobtained

_by

_giving

some

suitable value

to k. This ideais

sometimes

_expressed

by

saying that (i) are the

general co-ordinates of

any

point ofthe linejoining

P(x

ly

y

lt

zj and

Exercises

/I. Findtheco-ordinates of the pointswhichdivide the line _joining the

points(2, 4, 3),

(4,

5, 6)intheratios

(t) (1 : -4) and (w) (2 : 1).

[An*, (i) (4, -7,6) ; (n) (-2,2, -3).

2.

A

(3, 2, 0),

B

(6, 3, 2),

C

(-9, 6, -3) are three points forming a

triangle.

AD

the bisector of the angle

BAC,

meets

BG

at D. Find the

co-ordinates ofD. i

An

'

H)-3. Findthe ratioinwhichthe linejoiningthe_points

(2, 4, 5), (3,5, -4)

\a_{divided by}the _YZ-plane.

The

general co-ordinates ofanypointontheline_{joining thegivenpoints are}

\l

+

k ' 1

+

fc

'

1-ffc /

'"

This_pointwilllieon the

YZ

plane,if, and only if, its x co-ordinate is

zero,i.e.,

Hence

therequiredratio= 2: 3. _Putting

&=

_{2/3 in(t),}

wo

seethat the

point of intersectionis(0,2,23).

/_{4. Findthe}ratio inwhichthe

XY

-plane dividesthejoin of

(-3,4, -8) and (5, -6,4).

Also obtain_{the point of intersection of}thelinewith_{the plane.}

[Ana. 2 ;(7/3, -8/3,0).

5.

The

three points X(0, 0, 0), B(2, 3, 3) C( 2,3, 3), are collinear.

Findinwhatratioeach_{point divides}the_segmentjoiningtheother two.

[Ana.

ABIBC=,

BC/CA^2,

CA/AB^l.

6.

Show

thatthe_followingsets_{of points are}collinear:

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

()

(5, 4, 2), (6, 2, -1),(8, -2, -7).

7. Find the ratios in which the join ofthe points(3, 2, 1), (1, 3, 2) is

divided _bythelocus of the_equation

3*2--722/2-j-12822^3. [Ana.

-2

:1 ;1 : -2.

8. -4(4, 8, 12), J5(2, 4, 6), C(3, 5, 4), Z>(5, 8, 5)are the four points; show

that thelines

AB

and

CD

intersect.

9.

Show

that the _point (1, 1,2), is

common

to thelineswhichjoin

(6,

-7,

0) to (16, -19, -4) and(0, 3, -6)to (2, -5,10).

10.

Show

thattheco-ordinates of_anypointon the _plane determined by

thethree points(xlty^tz), (x2,j/2,z2),and(#3,7/3,z 3),

may

be expressedinthe

form . ^ l+m-\-n ~~'

l+m+n

'

l+m+n

) *

11.

Show

thatthecentroid ofthe _triangle whoso verticesare (xr, yr, zr) ;

r=l,2, 3, is

/s v

1*4. Tetrahedron.

Tetrahedron

is a _figure

bounded

_by

four

planes. Ithasfour vertices, each vertexarising as a point of

inter-section ofthreeof the four planes. Ithas six edges;each edge arising

as theline ofintersection of

two

of the _{four planes.} (

4

_C

2

=6).

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 thefour triangles,

ABC,

BCD,CAD,ABD-,

the four vertices are the points

A, B,

C,

D

and

thesix _edges arethelines

DIRECTION COSINES

OP

A

LINE 7

The

two

_edges

AB,

CD

joiningseparately the points,

A,

B

and

C,

D

are called a _pair of _{opposite edges. Similarly}

BC

9

AD

and CA,

BD

are the

two

other_{pairs of oppositeedges.}

Exercises

1.

The

four lines drawn from the vertices of _any tetrahedrontothe

centroidsoftheopposite facesmeet

m

a pointwhichis at three-fourths of the

distancefromeach vertexto the_oppositeface.

2.

Show

thatthethroelinesjoiningthe mid-points of opposite edges ofa

tetrahedronmoot inapoint.

1-5.

_Angle

between

two

lines.

The

_meaning

of the _angle

between

two

intersecting, i.e., coplanar lines, is _already

known

to the

student.

We

now

give the definitionof the angle

between

two

non-coplanarlines, also

sometimes

called

skew

lines.

Def.

The

_angle

between two

non-coplanar, i.e., non-intersecting

linesis _{the angle}

between two

_{intersecting lines}

drawn

from any

_point

parallelto each of the givenlines.

Note1.

To

justifythedefinitionofangle betweentwo non-coplanarlines,

as _{given above,} it is _{necessary to} showthat this_angleis_independentofthe

position ofthepointthrough which theparallel lines are drawn, but here

wo

simplyassumethis result.

Note2.

The

_angles between a givenlineandthe co-ordinate axes are

the_angleswhichthelinedrawnthrough the origin parallel to the given lino

makeswith theaxes.

^

1'6. Direction cosines of aline. Ifa, p,

J

be the angles

which

any

line

makes

with the_{positive directions} ofthe axes, thencos a,

cos _(3, cos

_y

are calledthe direction cosines_{of the given} line

and

are

generally

denoted

by

I,

m, n

respectively.

Ex.

What

arethedirection cosines of theaxesof co-ordinates?

[Ans. 1, 0,0;0, 1,0; 0,0, 1.

/

1*61.

A

useful relation. // be the _origin

and

_(xy

y

y z) the

co-ordinates ofa point P, then

xlr,

ymr,

z=nr

y

where Z,

m,

n

arethe direction cosines of

OP

and

r,is thelength of

OP.

Through

P

draw

PLJ_#-axis

so

that

OL=x.

From

the rt. _angled

triangle

OLP,

we

have

X

. i i.e.,

=6

or

x=lr.

Similarly

we have

2/=mr,

z=nr.

Fig. 6

^1*7.

Relation between direction cosines. _//

_{l,m and}

n

are the

direction cosines of

any

line, then

GEOMETRY

Let

OP

be

drawn

_through

the _{originparallel}to_{the given} line so

that Z,

m

t

n

are the cosinesofthe angles

which

OP

makes

with

OX

,

07,

OZ

respectively. (ReferFig. 6)

Let

(x, y, z) be the co-ordinatesof

any

point

P

on

this line.

Let

OP=r.

x=lr,

y=mr,

znr.

Squaring and

adding,

we

obtain

But

!2

+m

2

+n

2

=l.

Cor. If_a, 6,cbethree

numbers

_{proportional to}theactual direction

cosines Z,ra,

n

ofaline,

we

have

V

=

T

=

T

=

V(0

2

+&

a

+c

8

)

==4

~~~

XTf

where

the

same

_sign, positive or negative, is to

be

_{chosen throughout.}

*+ Direction Ratios.

From

above,

we

seethat a setofthree

numbers

which

are _proportionalto the actualdirection cosines are sufficient to

specify thedirectionof a line.

Such

numbers

are called the direction

ratios.

Thus

if a, 6, cbe thedirection ratios of a line, its direction

cosines are

Note. It is _easy to see that if a line

OP

_{^through theorigin} makes

anglesa, P,Ywith

OX,

Y, OZ, then the line

OP

obtained by producing

OP

Fig. 7

backwards through will

make

_angles TT a, TT

3, TC _{Y with}

OX,

OY,

OZ.

Thusif

cosa=Z, cos(3=w, cos

_Y=W

are the direction cosines ofOP, then

COS(TC a)=. Z, COS(TT 3)

=

m,COS(TC Y) n

PROJECTION OF

A

POlNf

ON A

Thusif

we

_{ignore the}twosenses ofaline,

we

can think of the direction

cosines /, m, n or /, m, ndetermining thedirection ofoneandthe

same

line. This_{explains the}ambiguityin_signobtainedabove.

Note. Thestudent_{should always}

make

a distinction between direction

cosinesanddirectionratios. Itis_only

when

/,?//,n are direction cosines, that

we

havethe relation

Exercises

1. 6,2, ,3are proportionaltothedirection cosines of a line.

What

are

theiractualvalues ? [Ana. (6/7,2/7,3/7).

2.

What

are the direction cosines oflines equally inclined to the axes?

How

_many

suchlinesare there? [Ana. (l/\/3, l/<\/3, dbl/V3

)J 4

-3.

The

co-ordinates ofa_point

P

are(3, 12, 4). Find thedirection cosines

ofthelineOP. [Ana. (3/13, 12/13, 4/13).

4. Thedirection cosinesI,m,n, oftwolinesare_{connected by}therelations

...($) Q. _...(ii)

Findthem?

Eliminatingn between(i) and(ii) ;

we

get

or

This equationgivestwovalues of l/mandhencethere are two lines.

The

tworoots of_(m)are 1 and

j.

IfIi,m>i,niand1%,

W

2 n2^e^nedirection cosines oftwolines,

we

have

mi

in2 2

Alsov

and . _J2

+

2

+

2

=0

or .

+

1

+

.

=

0, ^ ' ' * .

__

_

_

"T

-2

V6' ""Ve'

mi

~V^

ni V6'

5. Thedirection cosines oftwolinosaredeterminedbytherelations

(i)

/-5m+3n=0,

7/ 2 -f

Sw^-Sn^O

; find

them

? rx ,.v 1 2 3

112

[Ana. (*) -TT-T,

-

-, -7T7 7^' ~7^' "7^

\/14 \/14

v!4

-yb -yo -yO

, .x 1 3 4 1 2 3

1 '

V26

V

26

_V26

_V

14 _A/14

_V^

4

1*8. _Projection

on

aStraightline.

1*81. _Projection of a_point

on

aline.

The

footof the

10

ANALYTICAL

orthogonal projection (orsimply projection for the purpose of this

book) of the point

on

the line

and

is the

same

point -where

the _plane

through

the given point

and

perpendicular to the given line

meets

the

line.

Thus

in Fig. 1,

page

1,

A

is the

projec-_

-

_^

tion of

P

on

Z-axis ; also

B

and

C

are the

B

P

c

projectionsof

P

on

F-axis

and

Z-axis

respec-Fig. 8.

tively.

1-82. Projection ofa segment of a line on another line.

The

projectionof a

segment

AB

of a line

_{on any}

line

CD

is the

segment

A'B'

of

CD

where

A', B' are_{the projections} of A,

B

respectively

on

the line

CD.

Clearly

A'B'

is the _intercept

made

on

CD

_by

_planes

through

A,

B

each perpendicular to

CD.

Ex.

The

co-ordinates of a point

P

are (x, y, z).

What

are the projections

of

OP

ontheco-ordinateaxes ? [Ana. x, y,z.

Theorem.

The

_{projection of}a_given segment

AB

ofaline

on

any

line

CD

is

AB

cos 0, where is the_angle between

AB

and

CD.

Let

the _planes

_through

A

and

B

_{perpendicular} to theline

CD

meet

it in A',B' respectively so that

A'B'

isthe _projection of

AB.

Through

A

draw

a line

AP

\\

CD

to

meet

the plane

through

Bat

P.

AP

II

CD.

Now,

Also

BP

liesin the _plane

which

is

Hence

=

ABco*0

D

Fig. 9

Clearly

A'B'

PA

isa_{rectangle so} that

we

have

AP=A'B'.

Hence

A'B

f

=

AB

cos 6.

Cor. Direction cosines of the joinof

two

_points.

PROJECTION OF

A

BROKEN

LIKE 11

Let

L

9

M

be

thefeet ofthe perpendiculars

drawn

from

P,

Q

to

the_{JC-axis respectively so} that

OL=X

Projectionof

PQ

on X-axis=ZJlf

Also if I,

m, n

be thedirection cosines of

PQ,

the projectionof

PQ

on

_{X-a,xis=l.PQ.}

LPQ=x

2 Xi.

Similarly projecting

PQ

on

7-axis

and

Z-axis,

we

get

_

==

I

m

~~

n

Thus

the,direction _{cosines of}theline_joiningthe two_points

fa, 2/i, *i)

and

(x2, 2/ 2, z2)

are proportionalto

xtxi,

2/2-2/i, *2 *!

Exercises

1. Findthodirection cosines ofthelinesjoiningthe points

() (4, 3, -5) and(-2, 1, -8). [Ana. (6/7, 2/7,3/7),

(n) (7,-5,9) and(5, -3, 8). [Ans. (2/3, -2/3, 1/3)

2.

Show

that the points(1, 2, 3), (2,3, 4),(0, 7, 10)are collmear.

3.

The

projections ofalineon the axesare 12, 4, 3. Find thelength and

the direction cosines of theline. [Ana. 13 ; (12/13, 4/13, 3/13)

1*83. _Projectionof a brokenline _{(consistingof several} continuous

segments). If

P

ly

P

2,

P

3,...,

P

n be

any

number

of points in space,

thenthe

sum

ofthe_{projections of}

PlPl> ^2^3)...>

^n-l^n

on any

lineis _equal to the_{projection of}

_PiP

_n

on

the

same

line.

Let

_&,&,&,

...

_,Qn

ft

be

the _projections of the points

~Uonated

_by

P

₁₉ P*> P.,...>

Pn

Mr

-

N-

Sreeka

on

_{the given}line.

Then

M.Sc.

(Maths)

(

Ci#2=P

roiectionof PI?*,

Q*Qa~

P*PZ>

and

so on.

Also

_QiQn=

_projectionof

_PiP

n.

As

Qi,

Q

2>

Qz

...>

Qn

H

cm

the

same

line

we

have, for all

relative positions of these points

on

theline, the relation

Qi

v/_l*84.

Projection of thejoin of

two

pointson a line.

To

show

that

the_{projection of}the line_joining

on

aline with direction cosines Z,

m,

n

is

Through

P,

Q

draw

_{planes parallel} to the co-ordinate _planes to

form

a rectangularparallelopiped

whose

one diagonal is

_PQ.

_(See

Fig.3,

Page

3).

Now

The

lines

_PA,

_AN,

NQ

_{are respectively}

parallel to#-axis, y-axis,

z-axis. Therefore, their _{respective projections}

on

the line with

direction cosines I,

m,

n

are

fa

Xj)l9 (ya 2/i)w, (32 zi)n

-As

the _projection of

_PQ

_{on any}

line is _equal to the

sum

ofthe

projections of

PA, AN,

NQ

on. thatline, therefore the required

pro-jection is

(xa

xjl+fya

yi)m+(z

2 zjn,

Exercises

1. A(6,3, 2), (5, 1, 4),C(3, -4, 7), >(0, 2, 5) are fourpoints. Find the

projections of

AB

on

CD

andof

CD

on /!#. [4/is. 13/7 ; 13/3.

2.

Show

by projection that if P, <2, ti,

S

are the points(6, 6, 0),

(1,

;-7,6), (3, 4, 4,) (2, -9,2) respectivelythen

P^

_[_#.

v/l*9. Angle between

two

lines.

To

find the _anglebetween lines

whosedirection cosines are (^, ral5

%)

and

(/ 2,

m

2,

n

2 ).

Let

OP

1?

OP

2, be lines

through

the originparalleltothe given

linessothat the cosines ofthe _angles

which

OP

l

and

OP

2

make

with

Z

Fig. 10

the axes are Z

x,

m

l9 HI

and

72,

m

t,

n

2, respectively

and

the angle

between

the givenlines is_{the angle}

between

OP\ and

OP%.

Let

this

angle be 0.

ANGLE BETWEEN

TWO

LINES 13

The

projection of the line

OP

2 joining

0(0,0, 0)

and

P

2(x2, y,, z2)

on

theline

OP\

whose

direction cosinesare

is _xt

Alsothis _projection is

OP

₂ cos0.

OP

t cos

O^

But

a?2

=Z

a

.0P

2,

y^m^.OP^

z2

=n

2

.OP

2. _(1-61)

OP

2 COS

0=(y

or cos

=

Ijlg

Second Method.

Suppose

OPr^

0P

2

=r

2.

Let

the co-ordinates of Pj,

P

2,

be

(a^,

y

1? 2^)

and

(a:2, /2, 2;2)

respectively.

Then

^1

=

^1,

2/1=^^1,

Zi^r^,

(1*61)

and

We

have

Also

from

Trigonometry,

we

have

PiPf^rf+rf-toft

costf _...()

Therefore,

from

(i)

and

(n),

we

obtain

r^+-Sva

cos

_0=PiP

₂2

i.e., COS

0=^24-^1

Cor. 1. Sin

and

tan0.

The

_expressions forsin

and

tan in

a convenient

form

are obtained as follows :

sin2

0=l

cos2

sin

0=

sin0

and

tan0=

-

B=== ---v7~7

----cos6 _2/^2

Cor. 2. If the direction cosines of

two

lines

be

_proportional to

1, &!,cl5

and

2, 52) C2' tnen their actual valuesare

I 2 _L

> -L- it o i T, 2 I >. 2\* -1

SOLID

GEOMETRY

cos

₀₌

~

The

_expression for tan e is of the

same

form whether

we

use

direction cosines or direction ratios.

Cor. 3. Conditionsfor_{perpendicularity}

_and

parallelism.

(i)

When

thegiven lines areperpendicular,

0=90

sothat cos

_0=0.'

This _gives

a1a2-fb1

b

2

+c

1c2

=0.

(ii)

When

the given lines areparallel,

0=0

so thatsin

_0=0.

This_gives

which

istrue_only

when

ajftj o261

=0,

&iC2

62^=0,

CjO, _6284=0,

-^-^

A.

aa b2 eg

This result is also otherwise _evident,for, the lines_{through the}

origin

drawn

parallel tothe parallel lines coincide _and,

therefore, their

direction cosines

must

be the

same

and

hencedirection ratios

propor-tional.

Exercises

1. Find the_anglesbetweenthelineswhosedirectionratiosare

'(t) 5, -12, 13 ;-3, 4, 5. _[Ans. cog_-l(j/65)

(ii)

1,1,2;V3-1,-V3-1,4.

[Ans. w/3.'

2.

Show

thatthe_anglebetweenthelineswhosedirection cosines aretriven

bytherelationsinEx.4,P. 9is TT.

fe

/

3. Find the direction cosines of the linewhichis_{perpendicularto the lines}

withdirection cosines_proportionalto (7, 2, 2), (0, 2,1).

Sol. If I, m, n be the direction cosinesof theline_{perpendicular}tothe

givenlines,

we

have

Z(0)+m(2)-f n(l)=0,i.e.,OZ-f

2m+n=0.

These_give == _!!L ==_!L. l 2 2 1 2

'-V[2

2

₊

"(-"lTa

"+2]

~T'

w

=

F'

n==T"'

4.

Show

that a line can be found_{perpendicular to}thethreelineswith

direction cosines proportional to(2, 1,5), (4, -2,2), (-6,4, -1). Hence show

thatifthese threelinesbeconcurrent,theyarealso _coplanar.

5.

^ li

t

m

v

nx;/₂,_^2*n₂ rethedirection cosines

of two _mutually

EXERCISES 15

Sol. Ifltin,n bethedirection refines of the requiredline,wo have

111

+

nvm-i-fnn-i 0, 112-f

mm

2-fnn2 =0.

Thesegive

where8isthe_anglebetweenthegivenlines. As

_{0=90, we}

havesin0=1.

Hence

theresult.

6. _{li, Wj,}nt and1

2,

m

2,n2arcthe directionratiosoftwointersectinglines.

Show

thatlinesthroughthe intersection of thesetwowithdirectionratios

Ii+kl2,nii+kmtyHI~\kn2

are coplanarwiththem;kbeingany

number

whatsoever.

(Showthat_theyallhavea

common

perpendiculardirection.)

7.

Show

that threeconcurrentlineswithdirection cosines

(Il9

m

ltwj), (12>

m

2,n2), (J3,wi8,n8)

are coplanarif,

h,

n₂

m

3,

=0.

8.

Show

that the_{join of points (1,2,} 3), (4, 5, 7)is_parallel tothe join of

tho points(-4,3, -6),(2, 9, 2).

9.

Show

thatthe_points

(4, 7, 8) ; (2, 3,4) ;(-1, -2, 1) ; (1, 2,5)

aretheverticesofa_{parallelogram.}

10.

Show

thatthe_points

(5, -1, 1), (7, -4, 7), (1,-6, 10), (-1, -3, 4)

are thoverticesofarhombus.

11.

Show

thatthepoints.

(0,4,1), (2,3, -1), (4,5,0), (2,6,2)

are thevertices ofa_square.

^12. -4(1, 8, 4), B(0, 11,4), (7(2, 3, 1) are throe pointsand

D

istho foot

of_{the perpendicular}from

A

on

BC

. Findtheco-ordinates ofD.

[Ans. (4, 5, 2).

13. Findthe pointinwhichthejoinof

(9,

4, 5) and (11, 0, 1) is

met

bythe perpendicularfromtheorigin. [Ans. (1,2, 2).

14.

_A(~

1,2, 3),B(5,0, 6), C'(0, 4, 1) are three points.

Show

that

the direction cosines of the bisectors of the _angle

BAG

_{are proportional}to

(25, 8, 5)and(-11, 20, 23).

[Hint. Find theco-ordinates of the pointswhich divide

BC

in theratio

AB

:_AC.}

< _15. Findthe

anglebetweenthelineswhosedirectioncosines _{are given by}

the_{equations 3J-f-ra-|-5w=0}and

6mn2nl-\-5lmQ.

[Ans. cos-1

_^

, 16.

Show

that the pair of lines whose direction cosines are given by

32w 4lri-\-mn

=

Q, Z-f2w-j-3/i=0 are perpendicular.

17. Show that the _Straight lines whose direction cosines _{are given by}the

equations

are perpendicularor parallelaccording as

16

Sol. Eliminating/,betweenthegivenrelations,

we

have

?/(&///4-c/?)2 , . _n -, -

_+vw

2 -r-w>na

=0

a"

or _(b*u+a*(_i)''*

+

_{2Hbc>nn-\-(c2u+a2w)n*=Q ...(i)}

If tho lines be _parallel, theirdueotion cosmosare_equalsothatthe two

valuesof_{mjn must}be_equal. Theconditionfor thisis

+--

+

---0.

a '

v

~

w

Again,ifI_l9

w

_1}rijand1

2,w<2 7?₂bethe direction cosines of the two lines.

then equation (i) giv es

?J?1

?2

7'?

l?/'2 C

n_l /l_z w

]??2 6

or

Similarlytheelimination of/?,gives, (orby symmetry)

For_{perpendicular}lines

Thusthe conditionfor_{perpendicularity}is

a*(v-\-w}

+

bz(w+u)

+

c2(?/

+

?;)

=

0.

18.

Show

thatthestraightlineswhosedirectioncosinesare_givenby

are perpendicularif

andparallelif

19. /i, mj, ni ;7₂,w?₂,n2arethe direction cosines of two concurrent lines.

Show

that the direction cosines _of the _{lines bisecting theangles between}themare

proportionalto

Sol. Let thelinesconcurat the_origin

O

andletOA,

OB

bethetwo lines.

Take _points A,

B

on the twolinessuchthat(X4=0J3=r, say. Alsotake a

Fig. 11

pointA'on

AO

produced suchthatAO=--OA'. Let(7,C"be the mid-points of

AB

and A'B.

Then

O(7, OO' are the_requiredbisectors.

The

result,now,

followsfromthefactthatthe co-ordinatesof A, B, A'

respectivelyare

EXERCISES 17

20. _// the edges of a rectangular parallelepiped are a,b, cshow thatthe

angles betweenthe_{fourdiagonals are givenby}

Sol. Take one of the vertices

O

oftheparallelopiped asoriginand the

three rectangular facesthroughit_{as tho three rectangular co-ordinate planes.}

(SeeFig. 7, Page1).

Let(M=a,

OB=b,

OC=c

ThelinesOP, AL,

BM,

ON

arethe_{four diagonals.}

The

co-ordinates of_{A, B,}

C

are(a, 0,0) ; (0, 6, 0) ; (0, 0,c).

L,

M,

N

are _(0,6,c) ; (a, 0, c) ; (a, 6, 0).

0,

Pare

(0,0,0) ; (a,b, c).

Direction cosines of

OP

are _-,%r-$>

-,*-* ~7^>'>

V%

a _V2>a

_V2

a

Direction cosines of

AL

are

"1-are 2,

-n

a b _ C^V are

-^

,

-^

,

The_{angle between}

OP

and

CN

ttherefore,is

Similarlythoanglebetween any oneofthesix _{pairs of diagonals} can be

found.

21.

A

linemakesanglesa, p, y,S with the four diagonalsofa cube; prove

that cos2_a

+

cos2 p-f cos2

y+cos2

5

=

4/3. (P.U.1932)

(Choose axesasinEx. 20above andsupposethat the direction cosines of

the givenlinoareI, m,n.)

22.

_0,A,B,

C, arefourpointsnotlyingin the same plane and such that

OAJ^Bd

and

_OB^JIA.

Provethat

_OC^AB.

Whatwell-knowntheorem doesthis

becomeiffourpoints arcco-planar?

The

resultofthis_example

_may

alsobestatedthus:

"// two _pairsofoppositeedges ofa tetrahedron be at right angles,thenso is

thethird."

Take as_{originand any}threemutually perpendicularlines through as co-ordinato axes.

Let(.r lf T/J,Zj), (,r2, 7/2 z2), (#3, 2/3, z3) be the co-ordinates of the points

A

yBy

C

_{respectively.}

As

_OA^BC,

we

have

As

OB_[_CA,

we

have

r₂₍r_3-#l)+2

Adding(?') and (iV),

we

obtain

which showsthat 0(7 \

AB.

23. If,ina tetrahedron

OABC,

thenits_{pairs ofoppositeedges}are atrightangles.

24. /j, mj,

%

and/2

W

2 n2aretvvodirections inclined at an angle 9, to

eachother.

Show

thatthedirection

_ 2 cosi9 ' 2 cos <p ' 2 cos J9

18

Show

thatthesedirection cosines aretheactual values.

25.

Show

thatthedirection equally inclined to the three mutually

per-pendiculardirections

lit W'i HI J/2> Z>W2 >^3 ???3>n3

is _given

_by

thedirection cosines

/26.

Show

thatthearea ofthetrianglewhosevortices are the origin and

the_{points(x^ yj,}Zj),and (.r2,2/2, 22)is

iVCC^a-Wi)

2

_^^!^-^!)

2

-!-^!^-^^)

2 ]. (B.U.1958) 27. /i,wilfwl;/₂ wj 2 ?_*2 J^3 7?? 3 ??₃

arethedirection cosines of threemutually perpendicularlinos ;showthat

*1,hyJ

3 ;Wt.7"2 W'3!7?1 ^?_{2 ^3}

arealsothe direction cosines of three _{mutually perpendicular} lines. Hence

CHAPTER

II

THE PLANE

^

2*1. General _{equation of}first _degree.

_Every

_{equation of}the_first

degree in x,

y

y z representsaplane.

The

most

generalequationofthe first _degree inx,y, z is

where

a, 6, c arenot all zero.

The

locusofthis_equation will_{be a plane} if _{every point of} the

line _joining

_any

two

_points

on

thelocus also lies

on

thelocus.

To show

this,

we

take

any two

points

on

the locus, so that

we

have-Oy ...(i) 0. ...(ii)

Multiplying (ii)

by

k

and

adding to (z),

we

get

The

relation _(iii)

shows

that the point

_

V 1+k

'

l+k

'

1-fk

is also

on

the locus. But, for different values ofk, these arethe

general co-ordinatesof

any

point

on

the line

PQ.

Thus

every point

on

the _straightline _joining

_any

two

_{arbitrary points}

on

the locus also

lies

on

the locus.

The

_{given equation,} therefore, represents aplane.

Hence

_{every equation} of the first _degree in x, y, zrepresents

a _plane.

Ex

Findthe_{co-ordinates of the points}wherethe plane

ax+by+cz+d

Q

meetsthethree co-ordinate axes.

** _2*2.

_Normal

form

ofthe_{equation of a plane.}

To

_findthe_equation

ofa plane intermsof p, the lengthofthe

_{normal from}

the origin to it

and

Z,

m,

n

the direction cosines of that

normal

; (p is to bealways

regardedpositive).

Let

OK

be

the

normal from

O

to _{the given plane} ;

K

being the

footof the normal.

Then

_OK=p

and

I,

m,

n

areitsdirection cosines.

Take any

point P(x,y, z)

on

theplane.

20

GEOMETRY

Therefore the _projection of

OP

on

Fig. 12

Also the_projectionof the line

OP

_joining

0(0, 0, 0)

and P(x

9 y, z),

on

theline

OK

whose

direction cosines are

I,

m, n

>

is

l(x-Q)

+

m(y~-0)+n(z-0)=lx+my+nz.

(1*84, p. 14)

Hence

_lx+my+nz=p.

This _{equation, being} satisfied

_by

the co-ordinates of

any

point

P(x, y, z)

on

the given plane, represents the plane

and

is

known

as

the

_{normal form}

of_{the equation of a plane.}

Cor.

The

_{equation of}

_any

_plane is _ofthe_{firstdegree in x, y,} z.

Thisisthe converse of the

_{theorem proved}

in 2'1.

Ex. Find the _equation of the _{piano containing} the lines through the

originwithdirection cosines proportionalto _(1, 2, 2) and(2, 3, 1).

[Ans. 4cc _5y 7z=0.

^2-3.

Transformation to the

Normal

form.

To

transform the Aquation

tothe

normal

form

As

these

two

_{equations represent}the

same

_plane,

we

have

p

m

n

Thus,

djp=\/(a

2

+b

2 -{-c

z

)

and

as p, according to our

con-vention, is to _{be always}

positive,

we

shalltake positiveor negative

signwith theradicalaccordingas,d, is_negative or_positive.

PARALLELISM

AND

pERpEtfoicuLARitY

OF

iWo

PLANES

21

Ifd be_negative,

we

have

_onlyto

_change

the _signs ofall these.

Thus

the

normal form

of_{the equation ax-\-by-\-cz-\-d=0}is

be

_p

8itive :

' ifd be ncgative'

^

2*31. Direction cosines of normal toaplane.

From

above

we

deduce

_{a very important} factthat thedirection cosinesof

normal

to

any

planeare proportional to theco-efficients of_{x, y, z} in its_equation

or, that the direction ratios of the

normal

to a plane are the

co-efficients ofx, y, z inits _equation.

Thus,

a, 6,c

are the direction ratios ofthe

normal

to the_plane

Ex. 1. Find thedirection cosmosofthenormalsto_{the planes}

(t)

a-3/

+

65

=

7.

()

.H-2/+22-l

=

0.

(Ans. (t) 2/7,-3/7,6/7, (it) 1/3, 2/3, 2/3.

Ex. 2.

Show

that thenormalstothe_planes

are inclinedtoeachotheratan _angle90.

V'

2*32.

Angle

betweentwo _planes.

_{Angle between}

two

_planes is

equal to the angle

between

the normals to

them

from any

point.

Thus

_{the angle}

between

the

two

_planes

ax-\-by-{-cz-\-dQ,

and

ax

-\-b$-\- cLz-\-dl

~

is _equalto the _angle

between

the lines withdirection ratios

a, 6,c,

a_i>bi> s\>

and

is, therefore_,

^

_2'33. Parallelism

and

perpendicularity of two planes.

Two

planes are parallelorperpendicular according as thenormals to

them

are parallel or perpendicular.

Thus

the two_planes

ax+by+cz+d=0

and

ax

+ 1$

fc_Lz

₊

_di=Q

willbe_{parallel^ if}

a/a1

=b/b

1

=c/c

1 ;

and

will _{be perpendicular, if}

aa1

+bb

1

+cc

1

=0.

Exercises

/I. Find theanglesbetweentheplanes

(l)

2x-y+2z

3,3a;+62/-f22=4. [Ana. CO8~* _(4/21).

(ft) 2*~t/+z=6,

x+y+2z=7.

[Ana. w/3.

2.

Show

that the equations

Q, by-\-cz+p~Q,cz-\-ax-\-

q=0

representplanes respectively perpendicularto

_XY,

YZ,

ZX

planes.

3.

Show

that ax-}-l>y-\-cz-\-d=^Q represents plane.s, perpendicular

respec-tively to'YZ,

ZX,

XY

planes,if a,6,cseparately vanish. (SimilartoEx.2).

4.

Show

that the_piano

t

T-f22/-3z-f-4

=

is_{perpendicular} toeachof_{the planes}

*

2'4. Determinationof a_{plane under given conditions.}

The

_general

equationax-\-by-\-cz-\-d= of a plane contains three arbitrary

cons-tants (ratios of the co-efficients a, b,c, d) and, therefore, a plane can

be found

to_satisfy three conditions each _giving rise to _only one

relation

between

the constants.

The

three constants can then be

determined

from

_{the three resulting} relations.

We

_give below a fewsetsofconditions

which

_{determine a plane}:

(i) passing

through

three iion-collinear_points ;

(ii) passing

through

twogivenpoints

and

perpendicularto a given

plane ;

(in) passing

through

a given point

and

perpendicularto _{two given}

planes.

^

2'41. _Intercept

form

of the _equation of a plane.

To

findthe

equation ofa plane in terms ofthe intercepts a, b3 c which it

makes on

the axes.

Let

_{the equation of}the _{plane be}

Ax+By+Cz+D=Q.

...(1)

The

co-ordinates of the _point in

which

this _{plane meets the}

X-axis are _given to be (a, 0, 0). Substituting these in _{equation (1),}

we

obtain

A

1

=

-.

Similarly

-JL=

L

~.-?_=_L

D

~b ;

D

T"

The

equation (1) can bere-written as

A

B

C

.

-^"-fl"--^

1'

so that, after substitution,

we

obtain

A

₊

_{+_*=!,}

a b c 5

as _{the required equation of the}plane.

Note.

The

fact that aplanemakesinterceptsa,6, c, on thethreeaxesis

equivalent to thestatementthatit_{passes through} the three points (a, 0, 0),

(0, b,0), (0, 0,c),sothatwhat

we

havereallydonehereistodetermine thethree

ratiosoftheco-efficients in(1) inorder that thesame

may

pass through these

*>LANE

THkOUGH

THkEfi POINTS

Ex. 1. Find the _{intercepts of theplane 2x} 3y -f4z=12onthe co-ordinate

axes. [Ans. 6, 4, 3.

Ex. 2.

A

piano meets the co-ordinate axes at A, B,

C

such that the

eentroid ofthetriangle

ABC

isthe_{point (a,}b,c); showthat the equation of the

planeis_#/-f

#/6-fz/c=3.

Ex. 3. Prove that a variable plane whichmovessothat the

sum

of the

reciprocals ofits_interceptson the three co-ordinate axes is constant, _passes

through afixed_point.

2*42. Plane _through three _points.

To

find the _{equation of}the

plane passingthrough the three_{non-collinear points}

(*i, y\> *i), (**> y*> **}, (#3,2/3,

a)-Let

_{the required equation of the plane be}

As

the _{given points}lie

on

the _plane,

we

have

Eliminating a,l)

3 c,d

from

(i) (iv),

we

have

"<. y, *>

'

2/2,

which

is _{the required equation} of the_plane.

Note. In actual numerical exercises, thestudentwouldfinditmore

con-venienttofollow themethodofthefirstexorcise below.

Exercises

v

1. Findtheequationofthe_planethrough

P(2, 2, -1),C(3,4, 2), JR(7, 0, 6).

The

generalequationof aplanethroughP(2,2, 1) is

a

_(x2)

+

b(y 2)-f-c(-f-l)=0 (Refer4, 2-5,#.25) ...(t)

It will_passthrough

Q

and7?,if

5a-2&

₊

7c=0.

These_give

a b c

W=T

=

=ii

Substituting these valuesin (t),

we

have

asthe_{required equation.}

/12. Find the equation of the plane through the three _points

(1, 1,1),

(1, 1, 1),

(7,

3, 5)and showthatit isperpendicular to the

XZ

plane.