Table Of ContentDOME'? RISME
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SE 030 1435
ED1611 326
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Are, H. Glennc trd Others
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Analytic Geometry, 'Student Text., Paprt-1. Revisel
TITLE
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Edittolk.
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Stanford Univ. , Calit. School Mathematics -Study
INSTITUTION
Grtoup.
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NatiOnal Science Foundation, Washington, lii.V.,
SOONS A3EN,CY
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Algebra; *Analytic GeOmetry; Geometric Con,ceptss-
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DESCRIPTIRS
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Geometry: 'Graphs; Matheliatiss.Curriculum: Mathematioi
1Instructipia; Secondary Fduc4tioytSecondary.S9bool '
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Mathemattcs; *Textbooks
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*School Mathematics Study Group
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IDENTIFIER'S
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ABSTRACT
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This is part one of,,a threme-.peert
4c-itiol.Mathe,aatics
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Study Group (SIIS;) tertsbook. This part 6ontains tile first five
Analytic Geometry, (2) COOrtitnate's mid -the
°Ohapters 'including:
(1)
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(41 'Proo.f. by Aftaly,
(3) 'Vectors and Their Applications,
.1.4e,
Wards, ahd (5) Graphs and
eir Equatlions.,_ Mt
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ANALYTIt GEOMETRY
4
Student Text
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(Ervisid edition)
a.
W
4
PrePared hy-:
H. Glenn Are, Western Illinois Uthersity
a,
.William E. Briggs, University of Colorado
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Daniel
fonnec
Comiskey. The Taft Sehooh'Watertown,
John Dyer-Bennet, Carleoli College, Northfield,,Mi
Ewy, Fresno State College
Sandra Forsythe, Cubberley High School:Palo Alto, California
James H. Hood, San Jose Nigh School, San Jose, California
Max Kramer, San Jose State College, San Jose, California
Carol V. McCamman, Coolidge High School, Washington, D. C.
'William Wernick, Evander Childs High School, New York, Now York
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Trustees of thttleland Stanfo;d janior U?Ivasiry
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1963 and 1964 by The Board of
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reserved
All ri II
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States of America
Printed in the Uni
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Financial support for the School Mathematics Study Group has been
Science' F,undation.
provided by tke National
MALLOT . MC:
CUSHING
PNOTOLrrliGIMINT
ANN 4160.1t. MICHIGAN. UNITZP STATES OIr AIANNICA
AMON!
PREZFACE
Thia teXt aims at restoring what is, in a penSe, a "lost" subject.
There
widesprdia4 practice of irieludineanalytic geometry.tatarial in the colon.,
is
111B program; but when:this is-accomplished, Analytic geometry, as
a cOurse of
study, disppears and, wha).remains of it is the.iaxt immediately useful to&
study of calculus.
You will find a much more varied selection of topics in
this book than you would see in a cMu1us course.
In a booi devoted to the interplay betveen.algebra and geometry you would
expect,lo be.caildd uponfto exhibit considerable dexterity in algebraic mani
pulations,ap well ag to recall previous experiences with geometric figures and
You will not be disappointed.
theorele.
It is alio assix,med -that yb)1 know the
elementary:notions of trigonometry.
A deliberate effort was.made t; *tie this text to 'ProViouB MSG lbxts; so,
lyou will find the usual language of sets, ordered Pairs, nalmber properties,
This flavor is perhapg what
etc.,with wh?ch
you have had some acquaintance.
distinguisheg this book from others in the field. .,For example, the-t4atMe;It
IF
of coordinate systefflo in Chapter 2 depends upon the.po tulates of SMSG
geometry.
141,
The early chapters a re fundamental to every-
.Here)is one worsl,of advice.
Study them until they seeireto be old,f±iends; do nqt
thihg which follows.
hesitate to return to them later for a fresh look.
Another .thing you might
The related,ideas of vectors0-direction numl4rs, and parameters are
wat
ch.
Look for this
Used extensively, tb sitplify and unify the varidus topics.
feature.
.
The theoremg ana figures are nuMbered serially within each chapter; e.g.,
Theorem 8-3 is the*,third theorem of Chapter 8, Pigure 5-2 is the sectmd figure
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Ifan equation is to be referred too, it is aesigned
to appear in Chapter 5.
k
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The counting
a AUnting nuMber, which is then displayed in the left mahin.
Definitions are not numbered but may be foUnd
begins at one for each section.
by referring.to the Index.
The writers hope they have recreated the beauty of And,!Srtic Geometry in a
neW SMSG setting, and they ftiOther hope that you will enjoy andaprofit by the
adventure you are about to undertake.
Bldn'Yoyage.
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ciatimsY
ANALYTIC
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Whist Ib;Analytic Getmotry?
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1-2 Why Study Analytic Gemetry?
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-.Z.' COORDINATES AND THE Lin
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Linear Coordinate1SysteWs
2-1
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16
2.:2- Analytic Representations of Points and Sqbsets of a Line
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26
2,3 Coqrdinates it a Plane .
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Polar Coordinates
2-4
36
41
Lines in a Plane
2-5
2-6 Direction on a Line
57.
The Angle Between Two Lines; Parallel and Perpendicular
2-7
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:; Line4 s
2-8 .Normal arid Polar Forms of an Equation of a,Line
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.85.
2-9
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VECTORS AND THEIR APPLICATIONS
91
37
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3-1. WhyStudy Vectorsi
'91
DirectedLine Segments &mil Vectors
3-2
91
Sum snd'Estfference'bf Vectors; Scalar'Multiplicstion .
3-3
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Properties of-VectoreOperations'
3-4
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35 OLaraCterization of the Points on 4 Line
106
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Components
3-6
113'
Inner Product
121
3-7
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3-8 laws and 'Applicationi of the.Inner Product
127
3-9 Resfution of Vectors
131 411-
134
3q.0 Summary
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PROOFS BY ANALYTIC METHODS
4
4.
139
4
Introductabn
4-1
139
4-2 A-oofs Using Rectangular qoordinates
140
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Proofs Using Vectors
4-3
146
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154
4-4
Proofs Using Polar) Coordinates..
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Choice of WthOd of Proof
155
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GlIAPHS AND THEIR EQUATIONS
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Itetteduitan
5-1
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162
clad Graphs of Conditions
_5-2. Coiditions for Lol or Graphs
170
(5-1 Parametric Representation
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175
5-4 Parametric Equations of the Circle and,the Ellipse
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182
Parametric Equations of the Cyclold
5-5
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5:6 Parametric Equatiopi of a Straight Line
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194
Summary
5-7
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Chapter
.1
ANALYT= GEOMETRY
What Is Analytic Gebmet
1-1.
Geometry has beenlatudied
stenisticali,for over two thousand years.
EUclid's Elements, which'vas wri ten abaut 300 B.C., is perhapathe moot
There are undoubted.Wmany
ver published:
influential mathematics toxtbook
'traces or it to be found in the text you Ased in your high'sdhool courie.
studied.by Wh4
was are known as
Until.the llth century, geome
The postulates ealt with sucii. geometric notioqoa# point,
synthetic methods.
jine4 and angle, and littleior no use was made 6f numbers.
1h the Elements,
for example; line segments do not have lengths.
Then in the early part of thil7tlicentury,there pccurre p. the greatest
sinceEnolig.
advance in geometry
It was not the-work .or one man--auch
adiancs seldom, ifever, are.
Inbtead, it occvrrid When the "int.ellectual,
climate" was ready !or it. -Neveitheless,-there was one Man Whose,name is so
That
universally associeted with the new geometry that you ahould know it.
phligsoPherit.who lived
man vas Rent Descartes, a French mathematician and
tiat it
The essential .noVelty in .the new geometry wss
fram 1596 to 1650.
Thutilt brought together
used-algebrsic methods to solve geometric problems.
two subjects width until then had remained almost independent.
The,link between geometry and algebra is forged toe.coordiniite sysiems.
In essence, a coordinate system is a correspondence 'between the points of-same
and certain ordered sets of numbers. .(We ute quotation"marks because
II space"
the space m ay be a curve, or the surface of a sphere, or some other set of
points not UMIR14 th6ught of as a epace.) .YOu are already familiar with a
nuMber of different coordinate systema,'Somegetudied in earlier mathematics'
In element&ry
courses, others met with in other fields, suCh as geograwohy.
b; drajng two mutuelly
a plane
coordinatam into
algebra you introduced
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-perpendicular lines (axes) in the plane, choosing a positive direction on etch
and a unit length common ta both, and associating with eachopoint the ordered
pair,of Veal nuabers reprlsenting the directed distances of.the'point from t4e
The location of a point on the earth's surface it often.given tn
two axes.
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termn of latitude and 1 ngitude. An artilleryman sometimes locates'a target
by saying how far awAy.i\t ts, and'in What airectiop it lies with respect to
an
This
arbiti:ary fixed direction,established by betting up an aiming post.
is what is called a polar\coprdinate-systedlor-the plane.
Target
.
a
Aiming post
.
Artilleryman.
Fidire 1-1.
13'
on atright eircularcylinder could be identified by means of
P
(A point
a
Measure-pf thp angle 0
shbwn im
z, andthe
the directed distance
a
1
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4
Figure 1-2
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4
1-1
If, instead of one right circular cylinder, we considersall 8=4 cylinders
alth the same axis, pe can locate any Feint in space by giviing the iadips
r
of the cylinder On 'which it lies and 1'4
z-
coordinates on that
9-
cylinder.
The result is called a cylindrical cc
dinate.pystem for space.
A fly on a doughnut (a point on a torup) could ba localled
bY means of
the reasures (in 'degrees, radians,eilany other
ronvilaient Unit) of the angles
8
0
and
Shawn in the figure below.
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FigUre 1-3
The position of"an artificial.satellite at a certainm)ment could be #
specified by giving.its vortica;:distance from-the earthle suriace (or center)
and the latitude ind longitude of the'point of the e-orthts turface direct4.
4P
n
H the satellite.
:Figure 1-4
The resuleis called a sphe11.ca1 coordinate system for space.
Q.
3
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