Claude Irwin Palmer.

Analytic geometry, with introductory chapter on the calculus online

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ANALYTIC GEOMETRY






PUBLISHERS OF BOOKS F O R^

Coal Age v Electric Railway Journal

Electrical World v Engineering News -Record

American Machinist, v Ingenieria Internacional

Engineering'^ Mining Journal ^ Power

Chemical 6 Metallurgical Engineering

Electrical Merchandising



ANALYTIC GEOMETRY

WITH INTRODUCTORY CHAPTER ON THE

CALCULUS



BY

CLAUDE IRWIN PALMER

ASSOCIATE PROFESSOR OF MATHEMATICS,
ARMOUR INSTITUTE OF TECHNOLOGY



AND



WILLIAM CHARLES KRATHWOHL

ASSOCIATE PROFESSOR OF MATHEMATICS,
ARMOUR INSTITUTE OF TECHNOLOGY



FIRST EDITION



McGRAW-HILL BOOK COMPANY, INC.
NEW YORK: 370 SEVENTH AVENUE

LONDON: 6 & 8 BOUVERIE ST., E. C. 4
1921



COPYRIGHT, 1.921, BY THE
McGRAw-HiLL BOOK COMPANY, INC.



THE MAlr,IC PHKSS YORK PA






PREFACE



The object of this book is to present analytic geometry to
the student in as natural and simple a manner as possible
without losing mathematical rigor. The average student
thinks visually instead of abstractly, and it is for the average
student that this work has been written. It was prepared
primarily to meet the requirements in mathematics for the
second half of the first year at the Armour Institute of Tech-
nology. To make it adaptable to courses in other institutions
of learning certain topics not usually taught in an engineering
school have been added.

While it is useless to claim any great originality in treat-
ment or in the selection of subject matter, the methods and
illustrations have been thoroughly tested in the class room.
It is believed that the topics are so presented as to bring the
ideas within the grasp of students found in classes where
mathematics is a required subject. No attempt has been
made to be novel only; but the best ideas and treatment have
been used, no matter how often they have appeared in other
works on the subject.

The following points are to be especially noted:

(1) The great central idea is the passing from the geometric
to the analytic and vice versa. This idea is held consistently
throughout the book.

(2) In the beginning a broad foundation is laid in the
algebraic treatment of geometric ideas. Here the student
should acquire the analytic method if he is to make a success
of the course.

(3) Transformation of coordinates is given early and used
frequently throughout the book, not confined to a single
chapter as is so frequently the case. The same may be said
of polar coordinates.

v

M305O29



vi PREFACE

(4) Fundamental concepts are dealt with in an informal
as well as in a formal manner. The informal often fixes
and clarifies the ideas where the formal does not.

(5) Numerous illustrative examples are worked out in
order that the student may get a clear idea of the methods
to be used in the solution of problems.

(6) The conic sections are treated from the starting point
of the focus and directrix definition.

(7) Because of its great importance in engineering practice
the empirical equation is dealt with more completely than is
usual. This treatment has been made as elementary as
possible, but sufficiently comprehensive to enable one to
solve the average problem in empirical equations.

(8) The fundamental concepts of the calculus are presented
in a very concrete manner, and a much greater use then is
usual is made of the differential. The ideas are thus more
readily visualized than is possible otherwise. The applications
are mainly to tangents, normals, areas, and the discussion of
equations.

(9) The concluding chapter gives an adequate and careful
treatment of solid geometry so necessary in the study of the
calculus.

(10) The exercises are numerous, carefully graded, and
include many practical applications.

(11) In the introductory chapter are found various short
tables and formulas, and at the end are given four place
tables of logarithms and trigonometric functions.

The authors take this opportunity to express their indebted-
ness to their colleagues, Professors D. F. Campbell, H. R.
Phalen, and W. L. Miser, for their assistance in the preparation
of the text.

THE AUTHORS.
CHICAGO, ILL.,
May, 1921.



CONTENTS

CHAPTER I
INTRODUCTION

ART. PAGE

1. Introductory remarks : 1

2. Algebra and geometry united 1

3. Fundamental questions 1

4. Algebra 2

5. Trigonometry 3

6. Useful tables 5

CHAPTER II
GEOMETRIC FACTS EXPRESSED ANALYTICALLY, AND CONVERSELY

7. General statement . 8

8. Points as numbers, and conversely 8

9. The line segment 9

10. Addition and subtraction of line segments 10

11. Line segment between two points 11

12. Geometric addition and subtraction of line segments 11

13. Determination of a point in a plane 12

14. Coordinate axes 12

15. Plotting a point 13

16. Oblique cartesian coordinates . . 15

17. Notation . . . 15

18. Value of a line segment parallel to an axis 16

19 Distance between two points in rectangular coordinates ... 17

20. Internal and external division of a line segment 19

21. To find the coordinates of a point that divides a line segment in

a given ratio 20

22. Formulas for finding coordinates of point that divides a line
segment in a given ratio ' . 22

23. The angle between two lines 24

24. Inclination and slope of a line 25

25. Analytic expression for slope of a line 25

26. Formula for finding the slope of a line through two points . . 25

vii



viii CONTENTS

ART. PAGE

27. The tangent of the angle that one line makes with another in
terms of their slopes 26

28. Parallel and perpendicular lines 27

29. Location of points in a plane by polar coordinates 29

30. Relations between rectangular and polar coordinates 32

31. Changing from one system of axes to another 33

32. Translation of coordinate axes 33

33. Rotation of axes. Transformation to axes making an angle <p
with the original 34

34. Area of a triangle in rectangular coordinates 36

35. Area of any polygon 38

36. Analytic methods applied to the proofs of geometric theorems 39

CHAPTER III
Loci AND EQUATIONS

37. General statement 44

38. Constants and variables 44

39. The locus 45

40. The locus of an equation 45

41. Plotting an equation 46

42. The imaginary number in analytic geometry 47

43. Geometric facts from the equation 48

44. Intercepts 49

45. Symmetry, geometric properties 49

46. Symmetry, algebraic properties 50

47. Extent 51

84. Composite loci 53

49. Intersection of two curves 54

50. Equations of loci 55

51. Derivation of the equation of a locus 56

CHAPTER IV
THE STRAIGHT LINE AND THE GENERAL EQUATION OF THE FIRST DEGREE

52. Conditions determining a straight line 59

53. Point slope form of equation of the straight line 59

54. Lines parallel to the axes 60

55. Slope intercept form 61

56. Two point form 62

57. Intercept form 62



CONTENTS ix

ART. PAGE

58. Normal form 63

59. Linear equations 65

60. Plotting linear equations 65

61. Comparison of standard forms 66

62. Reduction of Ax + By + C = to the normal form .... 66

63. Distance from a point to a line 68

64. The bisectors of an angle 70

65. Systems of straight lines 71

66. Applications of systems of straight lines to problems 72

67. Loci through the intersection of two loci 75

68. Plotting by factoring . . . 77

69. Straight line in polar coordinates 78

70. Applications of the straight line 79

CHAPTER V
THE CIRCLE AND CERTAIN FORMS OF THE SECOND DEGREE EQUATION

71. Introduction . 86

72. Equation of circle in terms of center and radius 86

73. General equation of the circle 87

74. Special form of the general equation of the second degree . . 87

75. Equation of a circle satisfying three conditions 88

76. Systems of circles 92

77. Locus problems involving circles 94

78. Equation of a circle in polar coordinates 96

CHAPTER VI
THE PARABOLA AND CERTAIN FORMS OF THE SECOND DEGREE EQUATION

79. General statement 98

80. Conic sections * . . 98

81. Conies 99

82. The equation of the parabola 100

83. Shape of the parabola 101

84. Definitions 102

85. Parabola with axis on the y-axis 102

86. Equation of parabola when axes are translated 103

87. Equations of forms y 2 + Dx + Ey + F = and z 2 + Dx +

Ey + F = O 106

88. The quadratic function ax 2 + by + c 107

89. Equation simplified by translation of coordinate axes .... 107



X CONTENTS

ART. PAGE

90. Equation of a parabola when the coordinate axes are rotated 109

91. Equation of parabola in polar coordinates . Ill

92. Construction of a parabola 112

93. Parabolic arch 113

94. The path of a projectile 114

CHAPTER VII
THE ELLIPSE AND CERTAIN FORMS OF THE SECOND DEGREE EQUATION

95. The equation of the ellipse . 117

96. Shape of the ellipse 119

97. Definitions 120

98. Second focus and second directrix 120

99. Ellipse with major axis on the y-axis 121

100. Equation of ellipse when axes are translated 123

101. Equation of the form Ax 2 + Cy 2 + Dx + Ey + F = . . . 125

102. Equation of ellipse when axes are rotated 127

103. Equation of ellipse in polar coordinates 129

104. Construction of an ellipse 129

105. Uses of the ellipse 131

CHAPTER VIII

THE HYPERBOLA AND CERTAIN FORMS OF THE SECOND DEGREE

EQUATION

106. The equation of the hyperbola 134

107. Shape of the hyperbola 136

108. Definitions 137

109. Second focus and second directrix 137

110. Hyperbola with transverse axis on the y-axis 137

111. Asymptotes 139

112. Conjugate hyperbolas 141

113. Equilateral hyperbolas 142

114. Equation of hyperbola when axes are translated 143

115. Equation of the form Ax 2 + Cy 2 + Dx + Ey + F = . . . 144

116. Equation of hyperbola when axes are rotated 146

117. Equation of hyperbola in polar coordinates 147

118. Construction of an hyperbola 148

119. Uses of the hyperbola 150



CONTENTS xi



CHAPTER IX

OTHER Loci AND EQUATIONS
ART. PAGE

120. General statement 154

121. Summary for second degree equations 154

122. Suggestions for simplifying second degree equations 157

123. Parabolic type 158

124. Hyperbolic type 159

125. The cissoid of Diocles 160

126. Other algebraic equations 161

127. Exponential equations . 163

128. Applications 164

129. Logarithmic equations 165

130. The sine curve 167

131. Periodic functions 168

132. Period and amplitude of a function 169

133. Projection of a point having uniform circular motion. Simple
harmonic motion 170

134. Other applications of periodic functions 172

135. Exponential and periodic functions combined 172

136. Discussion of the equation 174

137. Loci of polar equations 175

138. Remarks on loci of polar equations 177

139. Spirals 178

140. Polar equation of a locus 178

-141. Parametric equations 180

142. The cycloid. . 182

143. The hypocycloid 183

144. The epicycloid 185

145. The involute of a circle 186



CHAPTER X

EMPIRICAL Loci AND EQUATIONS

146. General statement 188

147. Empirical curves 188

148. Experimental data .* 190

149. General forms of equations . \. 191

150. Straight line, y = mx + b 191

151. The method of least squares 193



xii CONTENTS

ART. PAGE

152. Parabolic type, y = cx n , n > 195

153. Hyperbolic type, y =cx n ,n<Q 197

154. Exponential type, y = ab x or y = ae kx 197

155. Probability curve 199

156. Logarithmic paper 200

157. Empirical formulas of the type y = a + bx + ex 2 + dx 3 + qx n 203

CHAPTER XI
POLES, POLARS, AND DIAMETERS

158. Harmonic ratio 206

159. Poles and polars 206

160. Properties of poles and polars 209

161. Diameters of an ellipse 210

162. Conjugate diameters of an ellipse 212

163. Diameters and conjugate diameters of an hyperbola .... 213

164. Diameters and conjugate diameters of a parabola 213

165. Diameters and conjugate diameters of the general conic . . .214

CHAPTER XII
ELEMENTS OF CALCULUS

166. Introductory remarks 216

167. Functions, variables, increments 216

168. Illustrations and definitions 220

169. Elementary theorems of limits 221

170. Derivatives 222

171. Tangents and normals 223

172. Differentiation by rules 225

173. The derivative when; (x) is x 226

174. The derivative when f(x) is c 226

175. The derivative of the sum of functions 226

176. The derivative of the product of two functions 227

177. The derivative of the product of a constant and a function 227

178. The derivative of the quotient of two functions 228

179. The derivative of the power of a function 228

180. Summary of formulas for algebraic functions 230

181. Examples of differentiation 231

182. Differentiation of implicit functions 233

183. Discussion of uses of derivative 235

184. Properties of a curve and its function . . . 235



CONTENTS xiii

ART. PAGE

185. Curves rising or falling, functions increasing or decreasing 236

186. Maximum and minimum 237

187. Concavity and point of inflection 239

188. Relations between increments 242

189. Differentials 243

190. Illustrations 243

191. The inverse of differentiation 246

192. Determination of the constant of integration 247

193. Methods of integrating 248

194. Trigonometric functions 250

195. Derivatives of sin u and cos u 250

196. Derivatives of other trigonometric functions 252

197. f sin udu and S cos udu 253

198. Derivative of log e u 254

199. Derivative of log a u " 255

200. Derivative of a u and e u 255

201. Derivative of 'U v 256

202. Illustrative examples 257

203. f^f e u du, and f a u du 258

CHAPTER XIII
SOLID ANALYTIC GEOMETRY

204. Introduction .... 261

205. Rectangular coordinates in space 261

206. Geometrical methods of finding the coordinates of a point in
space 263

207. Distance between two points 263

208. Coordinates of a point dividing a line segment in the ratio

ri to r 2 264

209. Orthogonal projections of line segments 266

210. Direction cosines of a line 267

211. Polar coordinates of a point 269

212. Spherical coordinates 270

213. Angle between two lines 271

214. Locus in space 274

215. Equations in one variable. Planes parallel to the axes .... 274

216. Equations in two variables. Cylindrical surfaces 274

217. Spheres 276

218. Surfaces of revolution 276

219. Equations of curves in space 278



Xiv CONTENTS

ART. PAGE

220. Sections of a surface by planes parallel to the coordinate planes. 279

221. Projections of curves on the coordinate planes 280

222. Surfaces in space 282

223. General equation of second degree 284

224. Ellipsoid 284

225. The hyperboloid of one sheet . 285

226. The hyperboloid of two sheets 287

227. Elliptic paraboloid 288

228. Hyperbolic paraboloid 289

229. Cone 290

230. Equation of a plane 292

231. General equation of a plane 292

232. Normal form of the equation of a plane 293

233. Reduction of the equation of a plane to the normal form . . . 293

234. Intercept form of the equation of a plane 294

235. The equation of a plane determined by three conditions . . 295

236. Angle between two planes 295

237. Distance from a point to a plane 296

238. Two plane equation of a straight line 298

239. Projection form of the equation of a straight line 298

240. Point direction form of the equation of a straight line,
symmetrical form , . . 299

241. Two point form of the equation of a straight line 300

SUMMARY OF FORMULAS 303

FOUR PLACE TABLE OF LOGARITHMS 308

TABLE OF TRIGONOMETRIC FUNCTIONS 310

ANSWERS 315

INDEX. 341



ANALYTIC GEOMETKY

CHAPTER I
INTRODUCTION

1. Introductory remarks. Although it is not always possi-
ble for a student to appreciate at the outset the content of a
subject, it is well, however, to consider the object of the study,
and to understand as far as possible its fundamental aims.

2. Algebra and geometry united. Analytic geometry, or
algebraic geometry, is a subject that unites algebra and geom-
etry in such a manner that each clarifies and helps the other.
Lagrange says: "As long as algebra and geometry travelled
separate paths their advance was slow and their applications
limited. But when these two sciences joined company, they
drew from each other fresh vitality and thenceforward marched
on at a rapid pace towards perfection. It is to Descartes 1
that we owe the application of algebra to geometry an appli-
cation which has furnished the key to the greatest discoveries
in all branches of mathematics."

3. Fundamental questions. The fundamental questions of
analytic geometry are three.

First, given a figure defined geometrically, to determine its
equation, or algebraic representation.

1 Rene* Descartes (1596-1650) was one of the most distinguished philos-
ophers. It was in pure mathematics, however, that he achieved the
greatest and most lasting results, especially by his invention of analytic
geometry. In developing this branch he had in mind the elucidation of
algebra by means of geometric intuition and concepts. He introduced
the present plan of representing known and unknown quantities, gave
standing to the present system of exponents, and set forth the well
known Descartes' Rule of Signs. His invention of analytic geometry
may be said to constitute the point of departure of modern mathematics.

1



2 ANALYTIC GEOMETRY [4

Second, given numbers or equations, to determine the geo-
metric figure corresponding to them.

Third, to study the relations that exist between the geo-
metric properties of a figure and the algebraic, or analytic,
properties of the equation.

To pursue the subject of analytic geometry successfully the
student should be familiar with plane and solid geometry, and
should know algebra through quadratic equations and plane
trigonometry.

While parts of analytic geometry can be applied at once to
the solution of various interesting and practical problems,
much of it is studied because it is used in more advanced
subjects in mathematics.

Some of the more frequently used facts of algebra and trig-
onometry are given here for convenience of reference.

4. Algebra. Quadratic equations. The roots of the quad-
ratic equation ax 2 + bx + c = are



b + \/b 2 4ac b v b 2 4ac

Tl = _ !_ i y and 7*2 = ^

a b c

TI + r% = > and r\r^ =

a a

These roots are

real and equal if b 2 4ac = 0,
real and unequal if b 2 4ac>0,
imaginary if b 2 4ac<0.

The expression b 2 4ac is called the discriminant of the
quadratic equation.
Logarithms.

(1) log MN = log M + log N.

(2) log (M + N) = log M - log N.

(3) log N n = n log N.

1 (5

= log N'.
n (6



5] INTRODUCTION 3

(7) a l SaN = N.

(8) log ^ = -log AT. (10) Iog 6 alog a 6 = 1.

1 (11) log, N = 2.302585 lo glo N.

N " = loga N ' (12) lo glo N = 0.43429 log, N.



The base e = 2.718281828459- .'.- 3.141592653589-
5. Trigonometry. Formulas.

(1) 2?r radians = 360, TT radians = 180.

1 80

(2) 1 radian = - - = 57.29578 - = 57 17' 44.8".

7T

(3) 1 = n = 0-0174533 - radiaas.

loll

(4) sin 2 6 + cos 2 = 1.

(5) 1 + tan 2 = sec 2 0.

(6) 1 + cot 2 = esc 2 0.

(7) sin = - 7, and esc =



_. uruva v^ov; v ir

csc sin

(8) cos = -, and sec = -

sec 0' cos

(9) tan = 7-2, and cot = -

cot tan

sin sec

(10) tan = = -

cos csc

/11N cos0 csc

(11) cot = -. 2 = ~

sin sec

(12) sin (a + |8) = sin a cos 5 + cos a sin 0.

(13) cos (a + #) = cos a cos j8 sin a sin /3.

(14) sin (a ft) = sin a cos j8 cos a sin 0.

(15) cos (a j8) = cos a cos /? + sin a sin /S.

tan a + tan

(16) tan (a + 0) =

1 tan a tan

/^^N / o\ tana tan

(17) tan (a - 0) = ~r, *

1 + tan a tan /3

(18) sin 20 = 2 sin cos 0.

(19) cos 20 = cos 2 - sin = 1 - 2 sin 2 = 2 cos 2 - 1.

2 tan



ANALYTIC GEOMETRY I 5



(21) sin *0 = J 1 C S g - (22) cos }0 = J^



(23) tan \9 ~ cos ' 1 ~ cos



+ cos 6 sin 6 1 + cos 0"

(24) sin + sin = 2 sin J( + #) cos (<* - 0).

(25) sin a - sin p = 2 cos J(a + 0) sin (a - /3).

(26) cos a + cos = 2 cos |(a + 0) cos i(a 0).

(27) cos a - cos ft = -2 sin \(a + 0) sin |(a - 0).

(28) sin a cos ]3 = | sin (a + 0) + J sin (a 0).

(29) cos a sin ft = \ sin (* + /*)-$ sin (a - 0).

(30) cos a cos = \ cos (a + |8) + J cos (a /3).

(31) sin a sin = - J cos (a + 0) + | cos (a 0).

(32) ^- = -A- = -A~ (Sine Law.)
sin a sin sm 7

(33) a 2 = 6 2 + c 2 - 26c cos a. (Cosine Law.)

(34) sin (|TT - 0) = cos 0.
cos (^TT 0) = sin 0.
tan (^TT 0) = cot 0.
cot (J?r 0) = tan 0.

(35) sin (|TT + 0) = cos 0.
cos (^7T + 0) = sin 0.
tan(i?r + 0) = -cot0.
cot (^TT + 0) = -tan0.

(36) sin ( TT - 0) = sin0.

COS ( 7T 0) = COS 0.

tan( TT - 0) = -tan0.
cot ( TT - 0) = -cot 0.

(37) sin ( TT + 0) = -sin0.

COS ( 7T + 0) = COS 0.

tan ( TT + 0) = tan 0.
cot ( TT + 0) = cot 0.

(38) sin (V - 0) = -cos 0.
cos (frr 0) = sin 0.
tan (|TT - 0) = cot 0.
COt (|7r - 0) = tan 0.



6]



INTRODUCTION



(39) sin (frr + 0) = -cos0.
cos (|7r + 0) = sin 6.
tan (|7r + 0) = -cot0.
cot (|7r + 0) = -tan0.

(40) sin (27T - 0) = -sin0.
cos (2ir 0) = cos 0.
tan(2?r - 0) = -tan0.
cot (2ir - 0) = -cot0.

(41) sin ( 0) = sin 0.
cos ( 0) = cos 0.
tan ( 0) = tan 0.
cot ( - 0) = -cot0.

6. Useful tables.



VALUES OF e x FROM x = TO x 4.9



X


0.0


0.1


0.2


0.3


0.4


0.5


0.6


0.7


0.8


0.9





1.00


1.11


1.22


1.35


1.49


1.65


1.82


2.01


2.23


2.46


1


2.72


3.00


3.32


3.67


4.06


4.48


4.95


5.47


6.05


6.69


2


7.39


8.17


9.03


9.97


11.0


12.2


13.5


14.9


16.4


18.2


3


20.1


22.2


24.5


27.1


30.0


33.1


36.6


40.4


44.7


49.4


4


54.6


60.3


66.7


73.7


81.5


90.0


99.5


109.9


121.5


134.3



VALUES OF e~ x FROM x = TO x 4.9



X


0.0


0.1


0.2


0.3


0.4


0.5


0.6


0.7


0.8


0.9





1.00


0.90


0.82


0.74


0.67


0.61


0.55


0.50


0.45


0.41


1


0.37


0.33


0.30


0.27


0.25


0.22


0.20


0.18


0.17


0.15


2


0.14


0.12


0.11


0.10


0.09


0.08


0.07


0.07


0.06


0.06


3


0.05


0.05


0.04


0.04


0.03


0.03


0.03


0.02


0.02


0.02


4


0.02


0.02


0.01


0.01


0.01


0.01


0.01


0.01


0.01


0.01



ANALYTIC GEOMETRY



o



EH




w



o



+

rH



rH fl







o
5



6] f^P 1 *^^ INTRODUCTION 7

TABLE OF FREQUENTLY USED TRIGONOMETRIC FUNCTIONS






in
radians


sin


cos


tan


cot


sec


CSC











1





00


1


CO


3Q


7T

6


1
2


V3
2


V3
3


V3


2\/3
3


2


45


7T

4


V2
2


V2
2


1


1


V2


V2


60


7T

3


V3


1
2


V3


V3


2


2V3


2


3


3


90


7T

2


1





00





oo


1


120


27T


V3


1


/3


V3


_2


2\/3




3


2


2


V


3




3


135


37T


V2


_v?


-1


-1


-V2


V2




4


2


2










150


57T


1
2


V3


V3


-V3


2\/3


2


6


2


3


3


180


TT





-1





00


-1


00


210


77T


1


V3


V3


\/3


2\/3


-2




6


2


2


3




3




225


57T


V2


-V?'


1


1


-V2


-V2




4


2


2










240


47T


A/3


1
2


V3


V3


-2


2\/3


3


2


3


3


270


37T

2


-1





oo





00


-1


300


57T


V3


1
2


-V3


V3


2


2\/3


3


2


3


3


315


TTT


_vj?


V2


-1


-1


V/2


-V2




4


2


2










330


UTT

6


1
~2


V3


V3


-V3


2V3


2


2


3


3


360


2ir





1





oo


1


00



CHAPTER II

GEOMETRIC FACTS EXPRESSED ANALYTICALLY, AND
CONVERSELY

7. General statement. Geometry deals with points, lines,
and figures composed of points and lines. Algebra deals with
numbers and algebraic statements composed of numbers, such
as the equation.

In order to study geometric relations by means of algebra,
and conversely, it is necessary to be able to represent points,
lines, and geometric figures by means of numbers and equa-
tions, and conversely. That is, it is necessary to be able to
translate from the language of geometry to that of algebra, and
conversely,

8. Points as numbers, and conversely. If a point moves
from A to B in a straight line, the point is said to generate
the line segment A B y that is, the line segment A B is the locus
of the point. If the point moves from B to A it generates

the line segment BA. It is con-

*? venient to consider A B and B A as

FlG j separate line segments having oppo-

site directions. The arrow is often
used to denote the positive direction.

Such line segments as A B and BA are called directed line
segments. The point from which the moving point starts is
called the initial point, and the point where it stops is called
the terminal point.

It is to be noted that a line segment is read by naming the
initial point first.

Let X'X be a straight line of indefinite length, and


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