• No results found

Calculus Multivariable 2nd Edition Blank & Krantz - Vector Calculus PDF

N/A
N/A
Protected

Academic year: 2021

Share "Calculus Multivariable 2nd Edition Blank & Krantz - Vector Calculus PDF"

Copied!
506
0
0

Loading.... (view fulltext now)

Full text

(1)
(2)
(3)

WILEY

PLUS

www.wileyplus.com

accessible, affordable,

active learning

WileyPLUS is an innovative, research-based, online environment for

effective teaching and learning.

WileyPLUS

•••

... motivates students with

confidence-boosting

feed back and proof of

progress, 24/7.

-;::;�

-::::r.."T:..-1•­

....

-. -.-. supports instructors with

reliable resources that

reinforce course goals

inside and outside of

the classroom .

(4)

PLUS

)

www.wileyplus.com

ALL THE HELP,

RESOURCES

, AND PERSONAL

SUPPORT

YOU AND YOUR STUDENTS NEED!

www.wileyplus.com/resources

r

r

1l

1st

C

A

LASS

•••

AND BEYOND!

2-Minute Tutorials and all of the resources you & your students need to get started.

QuickStart

Pre-loaded, ready-to-use assignments and presentations. Created by subject matter experts.

Student Partner Program

Student support from an experienced student user.

Technical Support 24/7 FAQs, online chat, and phone support.

www.wileyplus.com/support

�WILEY

Collaborate with your colleagues, find a mentor, attend virtual and live

events, and view resources.

www.WhereFacultyConnect.com

Your WileyPLUS Account Manager. Personal training and implementation support.

(5)

Calculus

BRIAN E. BLANK

STEVEN G. KRANTZ

Washington University in St. Louis

(6)

Senior Editorial Assistant Development Editor Marketing Manager Project Manager Production Manager Production Editor Design Director Senior Designer Senior Media Editor Media Specialist

Production Management Services Cover and Chapter Opening Photo Text and Cover Designer

Pamela Lashbrook Anne Scanlan-Rohrer Debi Doyle Laura Abrams Dorothy Sinclair Sandra Dumas Harry Nolan Maddy Lesure Melissa Edwards Lisa Sabatini

MPS Limited, a Macmillan Company © Walter Bibikow/JAl/Corbis Madelyn Lesure

This book was set in 10/12 TimesTen-Roman at MPS Limited, a Macmillan Company, and printed and bound by R. R. Donnelley (Jefferson City). The cover was printed by R. R. Donnelley (Jefferson City).

Founded in 1807. John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundatin of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specificatinos and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www. wiley.com/go/citizenship.

This book is printed on acid free paper. 8

Copyright © 2011, 2006 by John Wiley & Sons, Inc. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of

the United States, please contact your local representative. ISBN 13 978-0470-45359-9 (Multivariable)

ISBN 13 978-0470-45360-5 (Single & Multivariable) ISBN 13 978-0470-60198-3 (Single Variable) Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

(7)

A

pebble for Louis.

BEB

(8)
(9)

Preface ix

Supplementary Resources xiii Acknowledgments xiv

About the Authors xvi c H A p T E R 9 Vectors 721

Preview 721

1 Vectors in the Plane 722

2 Vectors in Three-Dimensional Space 732 3 The Dot Product and Applications 741 4 The Cross Product and Triple Product 753 5 Lines and Planes in Space 766

Summary of Key Topics 782 Review Exercises 785

Genesis & Development 9 787

C H A PT E R 1 o Vector-Valued Functions 191 Preview 791

1 Vector-Valued Functions-Limits, Derivatives, and Continuity 792 2 Velocity and Acceleration 803

3 Tangent Vectors and Arc Length 813 4 Curvature 824

5 Applications of Vector-Valued Functions to Motion 834 Summary of Key Topics 850

Review Exercises 853

Genesis & Development 10 856

C H A PT E R 1 1 Functions of Several Variables 861 Preview 861

1 Functions of Several Variables 863 2 Cylinders and Quadric Surfaces 874 3 Limits and Continuity 882

4 Partial Derivatives 890

5 Differentiability and the Chain Rule 900 6 Gradients and Directional Derivatives 913 7 Tangent Planes 921

8 Maximum-Minimum Problems 932

(10)

9 Lagrange Multipliers 946 Summary of Key Topics 957 Review Exercises 960

Genesis & Development 11 963

C H A PT E R 1 2

Multiple Integrals

967 Preview 967

1 Double Integrals Over Rectangular Regions 968 2 Integration Over More General Regions 976 3 Calculation of Volumes of Solids 984

4 Polar Coordinates 990

5 Integrating in Polar Coordinates 999 6 Triple Integrals 1014

7 Physical Applications 1020 8 Other Coordinate Systems 1029 Summary of Key Topics 1037 Review Exercises 1042

Genesis & Development 12 1045

CH APTER 1 3

Vector Calculus

1049 Preview 1049

1 Vector Fields 1050 2 Line Integrals 1061

3 Conservative Vector Fields and Path Independence 1071 4 Divergence, Gradient, and Curl 1085

5 Green's Theorem 1094 6 Surface Integrals 1104 7 Stokes's Theorem 1116

8 The Divergence Theorem 1131 Summary of Key Topics 1139 Review Exercises 1143

Genesis & Development 13 1146

Table of Integrals

T-1

Formulas from

Calculus: Single Variable T-14

Answers to Selected Exercises

A·1

(11)

Calculus is one of the milestones of human thought. In addition to its longstanding role as the gateway to science and engineering, calculus is now found in a diverse array of applications in business, economics, medicine, biology, and the social sciences. In today's technological world, in which more and more ideas are quan­ tified, knowledge of calculus has become essential to an increasingly broad cross­ section of the population.

Today's students, more than ever, comprise a highly heterogeneous group. Calculus students come from a wide variety of disciplines and backgrounds. Some study the subject because it is required, and others do so because it will widen their career options. Mathematics majors are going into law, medicine, genome research, the technology sector, government agencies, and many other professions. As the teaching and learning of calculus is rethought, we must keep our students' back­ grounds and futures in mind.

In our text, we seek to offer the best in current calculus teaching. Starting in the 1980s, a vigorous discussion began about the approaches to and the methods of teaching calculus. Although we have not abandoned the basic framework of calculus instruction that has resulted from decades of experience, we have incorporated a number of the newer ideas that have been developed in recent years. We have worked hard to address the needs of today's students, bringing together time-tested as well as innovative pedagogy and exposition. Our goal is to enhance the critical thinking skills of the students who use our text so that they may proceed successfully in whatever major or discipline that they ultimately choose to study.

Many resources are available to instructors and students today, from Web sites to interactive tutorials. A calculus textbook must be a tool that the instructor can use to augment and bolster his or her lectures, classroom activities, and resources. It must speak compellingly to students and enhance their classroom experience. It must be carefully written in the accepted language of mathematics but at a level that is appropriate for students who are still learning that language. It must be lively and inviting. It must have useful and fascinating applications. It will acquaint students with the history of calculus and with a sense of what mathematics is all about. It will teach its readers technique but also teach them concepts. It will show students how to discover and build their own ideas and viewpoints in a scientific subject. Particularly important in today's world is that it will illustrate ideas using computer modeling and calculation. We have made every effort to insure that ours is such a calculus book.

To attain this goal, we have focused on offering our readers the following:

• A writing style that is lucid and readable

• Motivation for important topics that is crisp and clean • Examples that showcase all key ideas

• Seamless links between theory and applications

• Applications from diverse disciplines, including biology, economics, physics, and

engineering

(12)

• Graphical interpretations that reinforce concepts

• Numerical investigations that make abstract ideas more concrete

Content

We present topics in a sequence that is fairly close to what has become a standard order for multivariable calculus. This volume refers to only ten formulas from Calculus: Single Variable. To facilitate the use of this book for students who own a different single variable text, we have collected these equations and presented them on p. T-14. In the outline that follows, we draw attention to several sections that may be regarded as optional. Chapter numbering follows that of our eight chapter single variable text.

Chapter 9, the first chapter of multivariable calculus, begins with a section that introduces vectors in the plane. It is followed by a section that covers the same material in space. After sections on the dot and cross product, Chapter 9 concludes with a comprehensive account of lines and planes in space.

Chapter 10 is devoted to the differential calculus and geometry of space curves. The final two sections are concerned with curvature and associated concepts, including applications to motion and derivations of Kepler's Laws. Because these two sections are not used later in the text, they may be omitted by instructors who need additional time for other topics.

The differential calculus of functions of two and three variables is taken up in Chapter 11. All topics in the standard curriculum are discussed. One less con­ ventional topic that we treat is the development of order 2 and order 3 Taylor polynomials for functions of two variables. We use these ideas in the discriminant test for saddle points and local extrema but nowhere else. It is therefore feasible to omit the discussion of multivariable Taylor polynomials.

Chapter 12 is devoted to multiple integrals and their applications. In a section that precedes cylindrical and spherical coordinates, we develop polar coordinates ab initio. This section may be omitted, of course, if polar coordinates have been introduced earlier in the calculus curriculum.

The concluding chapter on vector calculus, Chapter 13, covers vector fields, line and surface integrals, divergence, curl, flux, Green's Theorem, Stokes's The­ orem, and the Divergence Theorem.

Structural Elements

We start each chapter with a preview of the topics that will be covered. This short initial discussion gives an overview and provides motivation for the chapter. Each section of the chapter concludes with three or four Quick Quiz questions located before the exercises. Some of these questions are true/false tests of the theory. Most are quick checks of the basic computations of the section. The final section of every chapter is followed by a summary of the important formulas, theorems,

definitions, and concepts that have been learned. This end-of-chapter summary is,

in tum, followed by a large collection of review which are similar to the worked examples found in the chapter. Each chapter ends with a section called Genesis &

(13)

Preface xi Development, in which we discuss the history and evolution of the material of the chapter. We hope that students and instructors will find these supplementary dis­ cussions to be enlightening.

Occasionally within the prose, we remind students of concepts that have been learned earlier in the text. Sometimes we offer previews of material still to come. These discussions are tagged A Look Back or A Look Forward (and sometimes both).

Calculus instructors frequently offer their insights at the blackboard. We have included discussions of this nature in our text and have tagged them Insights.

Proofs

During the reviewing of our text, and after the first edition, we received every possible opinion concerning the issue of proofs-from the passionate Every proof must be included to the equally fervent No proof should be presented, to the see­ mingly cynical It does not matter because students will skip over them anyway. In fact, mathematicians reading research articles often do skip over proofs, returning later to those that are necessary for a deeper understanding of the material. Such an approach is often a good idea for the calculus student: Read the statement of a theorem, proceed immediately to the examples that illustrate how the theorem is used, and only then, when you know what the theorem is really saying, turn back to the proof (or sketch of a proof, because, in some cases, we have chosen to omit details that seem more likely to confuse than to enlighten, preferring instead to concentrate on a key, illuminating idea).

Exercises

There is a mantra among mathematicians that calculus is learned by doing, not by watching. Exercises therefore constitute a crucial component of a calculus book. We have divided our end-of-section exercises into three types: Problems for Practice, Further Theory and Practice, and Calculator/Computer Exercises. In general, exercises of the first type follow the worked examples of the text fairly closely. We have provided an ample supply, often organized into groups that are linked to particular examples. Instructors may easily choose from these for creating assignments. Students will find plenty of unassigned problems for additional practice, if needed.

The Further Theory and Practice exercises are intended as a supplement that the instructor may use as desired. Many of these are thought problems or open­ ended problems. In our own courses, we often have used them sparingly and sometimes not at all. These exercises are a mixed group. Computational exercises that have been placed in this subsection are not necessarily more difficult than those in the Problems for Practice exercises. They may have been excluded from that group because they do not closely follow a worked example. Or their solutions may involve techniques from earlier sections. Or, on occasion, they may indeed be challenging.

(14)

The Calculator/Computer exercises give the students (and the instructor) an opportunity to see how technology can help us to see and to perceive. These are problems for exploration, but they are problems with a point. Each one teaches a lesson.

Notation

Throughout our text, we use notation that is consistent with the requirements of technology. Because square brackets, brace brackets, and parentheses mean dif­ ferent things to computer algebra systems, we do not use them interchangeably. For example, we use only parentheses to group terms.

Without exception, we enclose all functional arguments in parentheses. Thus, we write sin(x) and not sin x. With this convention, an expression such as cos(x)2 is unambiguously defined: It must mean the square of cos(x); it cannot mean the cosine of (x)2 because such an interpretation would understand (x)2 to be the argument of the cosine, which is impossible given that (x)2 is not found inside parentheses. Our experience is that students quickly adjust to this notation because it is logical and adheres to strict, exceptionless rules.

Occasionally, we use exp(x) to denote the exponential function. That is, we sometimes write exp(x) instead of <?. Although this practice is common in the mathematical literature, it is infrequently found in calculus books. However, because students will need to code the exponential function as exp(x) in Matlab and in Maple, and as Exp[x] in Mathematica, we believe we should introduce them to what has become an important alternative notation. Our classroom experience has been that once we make students aware that exp(x) and c mean the same thing, there is no confusion.

The Second Edition

For this second edition, we have moved our coverage of inverse functions to Chapter 1. Logarithmic functions now closely follow the introduction of expo­ nential functions in Chapter 2. The remaining standard transcendental functions of the calculus curriculum now appear in Chapter 3. The topic of related rates has been moved to the chapter on applications of the derivative. We deleted a section entirely devoted to applications of the derivative to economics, but some of the material survives in the exercises.

We did not discuss the use of tables of integrals in the first edition, but we now give examples and exercises in Chapter 5, The Integral. Accordingly, we now provide an extensive Table of Integrals. In the chapter on techniques of integration, the two sections devoted to partial fractions were separated in the first edition but are now contiguous. In Chapter 7, Applications of the Integral, the discussion of density functions has been expanded. The section devoted to separable differential equations has been moved to this chapter, and a new section on linear differential equations has been added. In that chapter, we also have expanded our discussion of density functions. Our treatment of the Taylor series in the first edition occupied an

(15)

Preface xiii entire chapter and was much longer than the norm. We shortened this material so that it now makes up two sections of the chapter on infinite series. Nevertheless, with discussions of Newton's binomial series and applications to differential equations, our treatment remains thorough.

The second edition of this text retains all of the dynamic features of the first edition, but it builds in new features to make it timely and lively. It is an up-to-the­ moment book that incorporates the best features of traditional and present-day methodologies. It is a calculus book for today's students and today's calculus classroom. It will teach students calculus and imbue students with a respect for mathematical ideas and an appreciation for mathematical thought. It will show students that mathematics is an essential part of our lives and make them want to learn more.

Supplementary Resources

WileyPLUS is an innovative, research-based, online environment for effective teaching and learning. (To learn more about Wiley PLUS, visit www.wileyplus.com.)

What do students receive with WileyPLUS?

• A research-based design. WileyPLUS provides an online environment that

integrates relevant resources, including the entire digital textbook, in an easy-to­ navigate framework that helps students study more effectively.

• WileyPLUS adds structure by organizing textbook content into smaller, more

manageable "chunks."

• Related media, examples, and sample practice items reinforce the learning

objectives.

• Innovative features such as calendars, visual progress tracking, and self­

evaluation tools improve time management and strengthen areas of weakness.

• One-on-one engagement. With WileyPLUS for Blank/Krantz, Calculus znct edi­

tion, students receive 2417 access to resources that promote positive learning outcomes. Students engage with related examples (in various media) and sample practice items, including the following:

• Animations based on key illustrations in each chapter

• Algorithmically generated exercises in which students can click on a "help"

button for hints, as well as a large cache of extra exercises with final answers

• Measurable outcomes. Throughout each study session, students can assess their

progress and gain immediate feedback. WileyPLUS provides precise reporting of strengths and weaknesses, as well as individualized quizzes, so that students are confident they are spending their time on the right things. With WileyPLUS, students always know the exact outcome of their efforts.

What do instructors receive with WileyPLUS?

• Reliable resources. WileyPLUS provides reliable, customizable resources that

reinforce course goals inside and outside of the classroom as well as visibility into individual student progress. Precreated materials and activities help instructors optimize their time:

(16)

• Customizable course plan. WileyPLUS comes with a precreated course plan

designed by a subject matter expert uniquely for this course. Simple drag-and­ drop tools make it easy to assign the course plan as-is or modify it to reflect your course syllabus.

• Course materials and assessment content. The following content is provided. • Lecture Notes PowerPoint Slides

• Classroom Response System (Clicker) Questions • Instructor's Solutions Manual

• Gradable Reading Assignment Questions (embedded with online text) • Question Assignments (end-of-chapter problems coded algorithmically with

hints, links to text, whiteboard/show work feature, and instructor-controlled problem solving help)

• Gradebook. WileyPLUS provides instant access to reports on trends in class

performance, student use of course materials, and progress towards learning objectives, helping inform decisions and drive classroom discussions.

Powered by proven technology and built on a foundation of cognitive research, WileyPLUS has enriched the education of millions of students in more than 20 countries around the world.

For further information about available resources to accompany Blank/Krantz, Calculus 2nd edition, see www.wiley.com/college/blank.

Acknowledgments

Over the years of the development of this text, we have profited from the com­ ments of our colleagues around the country. We would particularly like to thank Chi Keung Chung, Dennis DeTurck, and Steve Desjardins for the insights and suggestions they have shared with us. To Chi Keung Chung, Roger Lipsett, and Don Hartig, we owe sincere thanks for preventing many erroneous answers from finding their way to the back of the book.

We would also like to take this opportunity to express our appreciation to all our reviewers for the contributions and advice that they offered. Their duties did not include rooting out all of our mistakes, but they found several. We are grateful to them for every error averted. We alone are responsible for those that remain. Debut Edition David Calvis, Baldwin-Wallace College

Reviewers Gunnar Carlsson, Stanford University Chi Keung Cheung, Boston College

Dennis DeTurck, University of Pennsylvania Bruce Edwards, University of Florida Saber Elyadi, Trinity University

David Ellis, San Francisco State University

Salvatrice Keating, Eastern Connecticut State University Jerold Marsden, California Institute of Technology Jack Mealy, Austin College

Harold Parks, Oregon State University Ronald Taylor, Berry College

(17)

Second Edition

Anthony Aidoo, Eastern Connecticut State University

Reviewers

Alvin Bayless, Northwestern University

Maegan Bos, St. Lawrence University Chi Keung Cheung, Boston College Shai Cohen, University of Toronto Randall Crist, Creighton University Joyati Debnath, Winona State University Steve Desjardins, University of Ottawa Bob Devaney, Boston University

Esther Vergara Diaz, Universite de Bretagne Occidentale Allen Donsig, University of Nebraska, Lincoln

Hans Engler, Georgetown University Mary Erb, Georgetown University Aurelian Gheondea, Bikent University

Klara Grodzinsky, Georgia Institute of Technology

Caixing Gu, California Polytechnic State University, San Luis Obispo Mowaffaq Hajja, Yarmouk University

Mohammad Hallat, University of South Carolina Aiken

Preface xv

Don Hartig, California Polytechnic State University, San Luis Obispo Ayse Gui Isikyer, Gebze Institute of Technology

Robert Keller, Loras College

M. Paul Latiolais, Portland State University Tim Lucas, Pepperdine University

Jie Miao, Arkansas State University

Aidan Naughton, University of St. Andrews Zbigniew Nitecki, Tufts University

Harold Parks, Oregon State University Elena Pavelescu, Rice University

Laura Taalman, James Madison University Daina Taimina, Cornell University

Jamal Tartir, Youngstown State University Alex Smith, University of Wisconsin Eau Claire Gerard Watts, Kings College, London

(18)

xvi

respected writers. Their extensive experience in consulting for a variety of pro­ fessions enables them to bring to this project diverse and motivational applications as well as realistic and practical uses of the computer.

Brian E. Blank Brian E. Blank was a calculus student of William 0. J. Moser and Kohur Gowri­ sankaran. He received his B.Sc. degree from McGill University in 1975 and Ph.D. from Cornell University in 1980. He has taught calculus at the University of Texas, the University of Maryland, and Washington University in St. Louis.

Steven G. Krantz Steven G. Krantz was born in San Francisco, California in 1951. He earned his Bachelor's degree from the University of California at Santa Cruz in 1971 and his Ph.D. from Princeton University in 1974. Krantz has taught at the University of California Los Angeles, Princeton, Penn State, and Washington University in St. Louis. He has served as chair of the latter department. Krantz serves on the edi­ torial board of six journals. He has directed 17 Ph.D. students and 9 Masters students. Krantz has been awarded the UCLA Alumni Foundation Distinguished Teaching Award, the Chauvenet Prize of the Mathematical Association of America, and the Beckenbach Book Award of the Mathematical Association of America. He is the author of 165 scholarly papers and 55 books.

(19)

P R E V

Vectors

E W In this chapter, we lay the geometric foundation for all of our coming work with functions of two or more variables. Those studies will require us to have a mathematical model of three-dimensional space. Just as we use a real number as the mathematical model of a point on a line, and an ordered pair of real numbers as the mathematical model of a point in the plane, so we use an ordered triple

(x,

y,

z)

as the mathematical model of a point in three-dimensional space.

Equations among these space variables describe curves and surfaces in space. In this chapter, we will study the simplest such objects, lines and planes, in detail. As an aid to our investigations, we introduce a new concept, the vector, that will be a vital tool in every chapter to follow.

A vector may be thought of as an arrow that has a length and a direction, but whose initial point is of no importance. If we reposition the arrow while preserving its direction, then it still represents the same vector. We will learn how to add vectors and how to perform a number of other algebraic operations with them. The interplay between the geometry and algebra of vectors is the perfect device for understanding the relationship between the geometry of lines and planes and their Cartesian equations.

The vector construction is ideal for capturing many concepts in both algebra and geometry. But the use of vectors extends to other sciences as well. Many fundamental quantities in physics are understood by means of their direction and magnitude. Force, velocity, and acceleration are examples. Using the length of a vector to represent magnitude, we can model these physical quantities by means of vectors.

(20)

9.1

Vectors in the Plane

For many purposes in calculus and physics, we need a concept that simultaneously contains the notions of direction and magnitude. For instance, force, velocity, and acceleration are all quantities that have both direction and magnitude. In this section, we will develop a mathematical tool, the

vector,

for handling these concepts.

l•JM@hi[.]:@

A line segment AB between two points

A

and B is said to be a

directed line segment

if one endpoint is considered to be the initial point of the line segment and the other endpoint the terminal point. We denote the directed line segment with initial point

A

and terminal point B by

AB.

The same pair of points determines a second directed line segment,

BA,

which is said to be

opposite

in direction to

AB

(see Figure

1).

x x x

_. Figure 1 Line segment AB, directed line segment

AB,

and directed line segment

BA

The directed line segment

AB

may be thought of as a straight path from

A

to B. Its direction is indicated by an arrow, as in Figure

1.

We often find it useful to parameterize

AB

so that its direction is "respected." In such a parameterization, the initial point of the interval of paramaterization corresponds to

A,

and the endpoint of the interval of parameterization corresponds to B. Example

1

demonstrates this technique.

� EXAM PL E 1 Suppose that

A

=

(5, 6)

and B

=

(9,

1

4)

. Find a para­ meterization x

= f(t),y =g(t),0� t�1ofAB

that respects its direction.

Solution We are to find functions

f

and

g

so that the point

(f(t), g(t))

traverses the line segment from

A

to B as

t

increases from

0

to

1.

From the initial point

A

to the terminal point B, the x-displacement is

9 - 5,

or 4. The y-displacement is

1

4 -6,

or

8.

We use these displacements together with the coordinates of

A

to define

f(t)

=

5 + 4

t

and

g(t)

=

6 +

8t.

Then, at the endpoints, we have

(f(O), g(O))

=

(5,6)

=A and

(f(l), g(l))

=

(5 + 4, 6 +

8)

=B. Thus x =

f(t),

y

=g(t), O�t� 1

parameterizes a curve that begins at

A

and ends at B. To see that this parameterized curve is a line segment, we use the equation x

= f (t) =

5 +

4t

to obtain

t

= (x

- 5)/4.

It follows that y

= g(t) =

6 +

8t =

6 +

8(x

- 5)/4

=

6 +

2(x

- 5),

or y

=

2x

- 4,

which is the equation of a line. Thus the �uations x =

5 + 4

t

, y =

6 +

8t, 0 � t � 1

parameterize the directed line segment

A!J.

..,.

(21)

_. Figure 2 Directed line seg­ ments with the same length and direction as

AB.

9.1 Vectors in the Plane 723

INSIGHT The calculation of Example

1

may be carried out with any two points

A

= (xo, Yo)

and B

= (x1, Y1).

The result is that the equations

x = xo + t(x1 - xo),

y

=Yo+ t(y1 - Yo) (0:::::; t:::::; 1)

(9.1.1)

-->

parameterize the directed line segment AB.

A directed line segment is determined by its initial point, direction, and length.

-Given a directed line segment AB, it is important to be able to construct directed

-line segments with the same length and direction as AB but with other initial points. The next example shows how this is done.

� EXAM PL E 2 Let A =

(5, 6)

and B =

(9, 14)

as in Example 1. Let 0 denote

the origin

(0, 0).

If C =

(3, -6)

and S =

(0, 10),

then find points D, P, and R so that the

---+ ---+ ;::;;t ---+

directed line segments, 0 P, CD and R.l have the same length and direction as AB.

-Solution The length and direction of AB are both determined by its x-displacement

9

-

5

=

4

and y-displacement

14 - 6

=

8.

If we add these

displacements to the coordinates of the initial points 0 and C, then we obtain the terminal points P =

(0 + 4, 0 + 8)

= (

4, 8)

and D =

(3 + 4, -6 + 8)

=

(7, 2).

If we

subtract these displacements from the coordinates of the terminal point S, then we

- -

-obtain the initial point R =

(0 - 4, 10

-

8)

=

(-4, 2).

In this way, OP, CD, and RS

.

-have the same x- and y-mcrements as AB and therefore the same length and direction (see Figure

2.)

-11111

Vectors

Imagine that you are exerting a force of constant magnitude to push a stalled

_. Figure 3

automobile (Figure

3).

If the street is straight, then the direction of the force F that you exert is also constant. An arrow may be used to represent the direction of the force F, as in Figure

3.

We can use the length of the arrow to represent the

magnitude of F. The initial point of the arrow can be used to represent the point of application of F. As the car moves, the position of the arrow changes, but the direction and length do not. We can therefore regard the force F as a directed line segment that (a) has a fixed direction, (b) has a fixed length, and (c) can be applied at any point. These considerations suggest creating a mathematical object that captures the direction and length of a directed line segment but which disregards its initial point. Such an object is called a vector.

Let A and B be points in the plane. The collection of all directed

-line segments having the same length and direction as AB is said to be a vector.

Every directed line segment in this collection is said to represent the vector.

_J

To help distinguish between vectors and ordinary numbers, we often refer to real numbers as scalars. In textbooks, boldface type such as v is frequently used to denote a vector. When rendered by hand, the vector vis often denoted by v.

Informally, we can think of a vector as a "floating" arrow whose initial point can be chosen in any convenient way. Figure

2

illustrates four representations of a single

vector. In practice, we often refer to a vector by one of the directed line segments that represent it. This can lead to no confusion and is often very helpful. Thus we

(22)

.A Figure 4 The components v1

and v2 of vector v are the same in

each directed line segment representation.

may refer to the vector that appears in Figure 2 as vector

AB,

vector

PQ,

vector

-

-ON, or vector RS. Even though these four directed line segments are all different, they all represent the same vector.

When a particular directed line segment P0P

is used to represent a vector

v

(

as in Figure

4),

the x-displacement v1

=

x1 - x0 and they-displacement v2

=

y1 -y0 do not depend on the choice of initial point P0• We can therefore use the notation

(

v1, v2

)

to unambiguously denote v. Thus

v =

(

v1, v2

)

. The quantities v1 and v2 are said to be the components or entries of

(

vi, v2

)

. Notice that, if P

=

(

x, y

)

and 0

=

-(0, 0),

then vector OP is equal to

(

x -

0,

y -

0)

or

(

x, y

)

. Thus every point P

=

(

x, y

)

gives rise to the vector

(

x, y

)

that is represented by the directed line segment from

-the origin to P

(

Figure

5).

The vector OP=

(

x, y

)

is sometimes called the position vector of P

=

(

x, y

)

.

� EXAM PL E 3 Let A

= (5, 6)

and B

= (9,

14

)

as in Examples 1 and 2. Let

v

-be the vector that is represented by AB. Write

v

in terms of its components. If

v

is

-represented by the directed line segment 0 P with the origin 0 as initial point, then what is the terminal point P?

Solution If

v =(a, b),

then the components

a

and

b

of

v

are given by

a= 9- 5 = 4

and

b = 14 - 6 = 8.

Thus

v = ( 4, 8).

The point P

= ( 4, 8)

is the terminal point of the directed line segment that begins at the origin 0 and that represents

v = (4, 8).

Ref er again to Figure 2. .,..

Vector Algebra

In this subsection, we introduce some algebraic operations that can be performed with vectors.

y

p = (x,y)

x

0

.A Figure 5 The position vector of point P =

(x, y)

The sum

v +

w of two vectors

v =

(

v1, v2

)

and w

=

(

w1, w2

)

is formed by adding the vectors componentwise:

v +

w

=

(

v1, v2

)

+

(

w1, w2

)

=

(

v1

+

w1, v2

+

wz

)

.

We can interpret vector addition geometrically by first drawing a directed line segment

AB

that represents

v

and then drawing a directed line segment

BC

that represents w. Notice

(

as shown in Figure

6)

that the initial point of the second directed line segment is the terminal point of the first. The sum

v +

w is then equal

____,. ____,. -=-+ ____,.

to vector

AC.

In other words,

AB + BC =AC.

Figure 7 shows that the sum

v +

w is the diagonal of the parallelogram determined by

v

and w. It follows that v

+

w

=

w

+ v,

as is also evident from the algebraic definition.

c

A

(23)

y 0

V1

---I

I

>+----

A V1

---+1

_. Figure Sa >..>O y _. Figure Sb >.. < 0 A _. Figure 9 ,\v2 x

9.1 Vectors in the Plane 725

� EXAMPLE 4 Add the vectors v =

(-3, 9)

and

w

=

(1,8).

Solution We have v

+ w

=

(-3+1, 9 + 8)

=

(-2, 17)

. <11111 Next we define the vector analogue of the scalar

0.

The zero vector 0 is the vector both of whose components are

0.

That is, 0 =

(0, 0).

The zero vector is the identity for vector addition. This means that, if v is any vector, then v

+

0 = 0

+

v = v.

If v =

(

v1, v2

)

is a vector, and >.is a real number, then we define the scalar multiplication of v by >. to be

>.v =

(

>.v1, >.vz

) .

Geometrically, we think of scalar multiplication as producing a vector that is "parallel" to v. One may use the idea of similar triangles to check this idea, as Figure

8

suggests. If>.>

0,

then >.v and v have the same direction. If>.<

0,

then >..v and v have opposite directions. These intuitive notions will be made more precise when a formal definition of direction is given later.

� EXAMPLE 5 If v =

(2, -1),

then calculate 2v and -3v.

Solution We have 2v =

(2

·

2, 2

·

(-1))

=

(4, -2)

and - 3v =

((-3)

·

2, (-3)

·

(-1))

=

(-6, 3)

.

...

Vector addition and scalar multiplication can be used together to define vector subtraction. If v and

w

are given vectors, then the expression v

- w

is interpreted to mean v

+ ((-l)w).

See Figure

9,

which features the parallelogram that is generated by v and

w.

This same parallelogram has been used in

ure

7

to visualize the sum v

+ w,

which is represented by the directed diagonal AC. In Figure

9,

notice that

---+

v

-w

is represented by the other directed diagonal, DB, of the parallelogram. To understand why, focus on the three sides of the upper-right triangle, which show

---+ ---+ ---+

that DB =DC

+

CB = v

+ ( -w)

= v

- w.

Furthermore, by concentrating on the three sides of the lower-left triangle of Figure

9,

we see that v

- w

is the vector that C we add to

w

to obtain v.

A simple calculation shows that subtraction of vectors is performed componentwise, just like addition of vectors. Thus if v =

(

vi. v2

)

and

w

=

(

w1, w2

)

, then

v -w =

(

v1, v2

)

+

(

- l

)(

w1, w2

)

=

(

v1, v2

)

+

(

-w1, -wz

)

=

(

v1 - w1, vz - wz

)

. � EXAM PL E 6 Calculate the difference v

- w

of the vectors v =

(-6, 12)

and

w

=

(19, -7).

(24)

The Length

(or Magnitude) of

a Vector

If

v = (v1, v2)

is a vector, then its length is defined to be

llvll

=

V

(

v1

)2 + (v2)2.

-This quantity is also called the magnitude of

v.

If AB is a directed line segment that represents

v,

then II

v

II is just the distance between the points A and B. The length of a nonzero vector is positive. The zero vector 0 is the only vector that has length equal to

0.

• • ---+ ---+

-EXAM PL E 7 Let 0

= (0, 0)

denote the ongm. Calculate

llPQ

II, II

OP

II, and

llOQll,

for

P

= (-9,6)

and

Q

= (2, 1).

- - ::;-h

Solution Because

PQ =

(2- (-9), 1 - 6) = (11, -5),

OP=

(-9, 6),

and

O�

=

(2, 1),

we have

llPQll,

=

J

112 + (-5)2 =

JI46,

II

oPll

=

V

(-9)2 + 62 =

y1Il7,

and

llOQll,

= V22+12 =

v's .

....

THEOREM 1 If vis a vector and

A.

is a scalar, then

11>.v11 =1>-111v11·

(9.1.2)

In words: The length of

A.

v

is the absolute value of

A.

times the length of v.

Proof. Equation

(9.1.2)

is easily derived from the definitions of length and scalar multiplication:

II .\v II = II

(

.\vi, .\v2

)

II =

V

(

.\v1

)2 + (.\ v2)2 = v'>!-

J

(

v1

)2 + (v2)2 = l>-111 v II·

• � EXAMPLE 8 Verify equation

(9.1.2)

for

v= (4, 12)

and

A.= -5.

Solution We calculate

l>-1 11 v II= l-5IV 42+122 = 5v'l60 = 2ov1o

and

II>. v II =

II

(-5) (4, 12)

II

=

II

(-20, -60) II =

V

(-20)2 + (-60)2 = 20

J

(-1)2 + (-3)2 = 20JIQ.

....

Unit Vectors and

If the length of

u

= (u1, u2)

is

1,

that is, if

II

u

II = 1,

then

u

is called a unit vector.

Directions

Observe that

u

=

(

u1, u2

)

is a unit vector if and only if the point

(

u1,

u2)

lies on the

unit circle (Figure

10).

A unit vector

u

therefore has the form

u

=(cos( a

)

, sin( a

))

(9.1.3)

for some angle a in

[0,27!").

Sometimes we refer to a unit vector

u

as a direction vector or simply a direction. For any nonzero vector

v =(vi. v2),

the vector

1

u

= wv

(9.1.4)

is a unit vector because, according to equation

(9.1.2),

llull

=II

11

!

11

v

ii

=

I

(25)

y

1

-1

• Figure 10 A unit vector

u =

(u1, Uz)

y

x 1

Q

=

(2, 1)

x -1

2

-1 • Figure 11

9.1 Vectors in the Plane 727

We refer to the direction vector

u

defined by equation (9.1.4) as the direction of v

and write

dir(v)

=

11

!

11 v (v-=!= 0).

(9

.

1.5)

We do not define the direction of the zero vector. By rearranging equation (9.1.5),

we see that every nonzero vector v can be expressed as the scalar multiplication of

the direction vector of v by the magnitude of v:

v

=

llv ll dir(v).

(9.1.6)

The right side of equation (9.1.6) is sometimes said to be the polar form of v.

� EXAMPLE 9

Let P = (-1, 2) and Q = (2, 1). What is the direction

u

of the

- .

vector v represented by PQ? What angle

a,

as given by formula (9. 1.3), does

u

make with the positive x-axis?

Solution

We calculate v=(2- (-1), 1-2)=(3,-1) and llvll =

J

32+(-1)2=

JIO. The unit vector

is the direction of v. Figure 11 illustrates both v and its direction

u.

The angle

a

that

u

=(cos(

a

)

,

sin(

a

)) makes with the positive x-axis is the value of

a

between

37r /2 and 27r such that cos(

a

) = 3/M and sin( a)= 1/JIO. That is, a= 27r

-arctan ( 113). With the aid of a calculator, we find that

a�

5.96 radians.

<Ill

Let v and w be nonzero vectors. We say that v and w have the

same direction if dir(v) = dir(w). We say that v and ware opposite in direction if

dir(v) = -dir(w). We say that v and ware parallel if either (a) v and w have the

same direction or (b) v and w are opposite in direction. Although the zero

vector 0 does not have a direction, it is conventional to say that 0 is parallel to

every vector.

Notice that nonzero vectors v and ware parallel if and only if dir(v) = ±dir(w).

Our next theorem provides us with a simple algebraic condition for recognizing

parallel vectors.

THEOREM 2

Vectors v and w are parallel if and only if at least one of the

following two equations holds: (a) v = 0 or (b) w = A.v for some scalar A..

Moreover, if v and ware both nonzero and w

=

A.v, then v and w have the same

direction if

0 <

A. and opposite directions if A.

< 0.

Proof.

We begin by supposing that equation (a) or equation (b) is true. If v = 0,

then v and w are parallel by convention. Now suppose that v-=!= 0 and w = .Xv for

some scalar .X. If .X =

0,

then w = 0, in which case v and w are parallel. If .X -=!=

0,

then

(26)

(O, 100)

(75,0)

the following three facts hold: neither

v

nor

w

is 0, both

llvll

and

llwll

are nonzero, and both

dir(v)

and

dir(w)

are defined. Consequently, we have

dir(w)

=

11�11

w

=

II

;

vii

(

>.

v)

=

(

l

I

) 11!11

v

=

(

l

I

)

dir(v)

=

±dir(v).

This equation tells us that

v

and

w

are parallel. Furthermore, because

A.

I

IA.I =

+ 1 when

A. >

0

and

A.

I

IA.I =

-1

when

A. <

0,

we see that

v

and

w

have the same direction if

A.>

0

and opposite directions if

A.<

0.

To prove the converse, we suppose that

v

and

w

are parallel and that

v

=I= 0. We must prove that

w

= A.v

for some scalar

A..

If

w

=

0, then

w

= A.v

for

A.=

0.

We may therefore assume that

w,

in addition to

v,

is nonzero. In this case, both

dir(v)

and

dir(w)

are defined. Moreover, because

v

and ware parallel, we have

dir(v)

=

±dir(w).

Finally, noting that

llvll

=I= 0

and using equation (9.1.5), we calculate

w

=

llwlldir(w)

=

± llwlldir(v)

=

±

1/1:1111

v

= >.v

for>.=

±llwll/llvll·

� EXAMPLE 1 0 For what value of

a

are the vectors

(a,

-1)

and

(3, 4)

parallel?

_. Figure 12

Solution For the two given nonzero vectors to be parallel, it is necessary and sufficient for

(3, 4) =A. (a,

-1)

for some scalar>.. This vector equation is equivalent to the two scalar equations,

A.a= 3

and

-A.= 4.

We obtain

a= 3/A.,

or

a= -3/4.

<Ill

An Application

to Physics

As discussed earlier,

force

is a quantity that is naturally described by the vector concept. If a force of magnitude F is applied in direction (cos(o:), sin(o:)), then the force is written as the vector (Fcos(o:), Fsin(o:)). The

principle of superposit ion

says that, if two forces act on a body, then the

resultant force

is obtained by adding the vectors corresponding to the two given forces.

� E X A M P L E 1 1 Two workers are each pulling on a rope attached to a dead tree stump. One pulls in the northerly direction with a force of

100

pounds and the other in the easterly direction with a force of 75 pounds. Compute the resultant force that is applied to the tree stump.

Solution Look at Figure

12.

The force vector for the first worker is

(0, 100)

and that for the second worker is (75,

0).

The resultant force is the sum of these, or (75,

100).

The resultant force vector is shown in Figure

12.

The associated directed line segment has magnitude (length) equal to (75

2

+

10a2)112

= 125.

<Ill

The Special Unit

It is common to let

i

denote the vector

(1, 0)

and j denote the vector

(0,

1), as shown

Vectors i and j

in Figure 13. If

v

=

(a,

b)

is any vector, then we may express

v

in terms of

i

and

j

as

follows:

v

= (a,b) = a

(l,

O

)

+

b(O,

1)

= ai

+

bj.

� EXAM PL E 1 2 Suppose that

v

= (3,

-5) and

w

= (2, 4).

Express

v, w,

and

v

+

w

in terms of

i

and

j.

Solution We have

v

= 3i-

5j and

w

= 2i

+

4j.

We can calculate

(27)

The Triangle Inequality

y 1 j = (0, 1) x i = (1, 0) 1 .A Figure 13 Q UIC K Q UIZ

9.1 Vectors in the Plane 729

Alternatively, we can express

v

and win terms of

i

and j, as we have already done, and add these expressions:

v +

w

= (3i- 5j) + ( 2i + 4j) = (3 + 2) i + (-5 + 4) j =Si -

j. "ill

A glance back at MBC in Figure

7

suggests that, if

v

and w are nonparallel vectors, then the length of

v +

w is never greater than the sum of the lengths of

v

and w. In other words,

llv + wll

:5

llvll + llwll

·

(9.1.7)

In fact, this inequality, known as the

Triangle Inequality,

is just a vector inter­ pretation of the familiar theorem in Euclidean geometry that states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Inequality

(9.1.7)

is also true when

v

and ware parallel, for, in this case, we may write one of the vectors, say w, as a scalar multiple of the other:

v

=A. w. We then use equation

(9.1.2)

together with the Triangle Inequality for scalars to obtain

llv + wll = llv +>.vii = II (1 +>. )vii = 11 + >.i llvll

:5

(1 + i>.I) llvll

= llvll + i>.1 llvll = llvll +II >.vii = llvll + llwll

· <411

� EXAMPLE 13 Verify the Triangle Inequality for the vectors

v= (-3, 4)

and w

= (8, 6).

Solution After calculating

llvll + llwll =

J(-3) 2 + 42 + V82+62=5+ 10=15,

we find that

llv + wll =II (5, 10) II=

villi �

11.18<15 = llvll + llwll

· <1111

We conclude this section by stating the distributive, associative, and commu­ tative laws for scalar multiplication and vector addition . If

u,

v,

ware vectors and>.,

µ are scalars then a.

v+w=w+v

b .

u

+ (v +

w)

=

(u

+ v) +

w c.

>.(µv) = (>.µ)v

d .

>.(v +

w)

= >.v +

>.w e.

( >. +

µ

)v = >. v +

µ

v

• ---+

1.

If

P= ( 2, - 1) ,

then for what pomt

Q

does

PQ

represent

(-3, - 2)?

2.

What is the magnitude of

(-3, 4)?

3.

What is the direction vector of

(12, -5)?

4.

If

(a, b)

is opposite in direction to

(-4, 3)

and has length

2,

then what is

a?

Answers

(28)

EXERCISES

Problems for Practice

In each of Exercises 1-8, sketch

PQ,

and write it in the form (a,b). 1.

P=

(5,8) 2.

P=

(1,1) 3.

P=

(0,0) 4.

P=

(1,1) 5. P= (-1,8) 6.

P=

(0,1) 7.

P=

(-5,6)

8. P=

(2,2) Q = (4,4) Q = (3,9) Q=(-5,1) Q

=

(2,2) Q=(4,-3) Q = (0,0) Q

=

(1,0) Q

=

(2,-2)

In each of Exercises 9-12, calculate the lengths of

PQ

and RS. Also determine whether these vectors are parallel.

9. p = (1, 2) Q = (2, 0) 10. p = (1, 3) Q = (3, 1) lL

P=

(3,1) Q= (- 6,4) 12.P=(l,1) Q=(-7,5) R = (3,6) R=(l,2) R= (7,-2) R = (1,0) S= (6,0) S= (6,3) S= (5,5) S= (17,-8) In each of Exercises 13-16, calculate the indicated vector given that v = (4,2) and w = (1, -3). For each exercise, draw a sketch of v, w, and the vector you have calculated.

13. 2v+w

14. (1/2)v + 3w 15. v - 2w 16. -3v+2w

In each of Exercises 17-20, two points

P

and Q are given. Determine (a) the vector v that is represented by

PQ,

(b) the length of v, ( c) the vector that has the same length as v but is in the opposite direction of v, (d) the direction vector of v, and (e) a unit vector that is in the opposite direction of v.

17. p = (5, -7) 18. p

=

(-7, 1) 19. p = (-3, 7) 20. p = (3,2) Q = (0,5) Q

=

(-14) Q = (1,-9) Q = (l,1)

In each of Exercises 21-24, vectors v and w are given. Express v, w, -4v, 3v - 2w, and 4v + 7j in terms of the unit vectors i and j. 2L v= (-7, 2) 22. v= (0,3) 23. v = (6, -2) 24. v = (1/3, 1) w= (-2, 9) w=(-5, 0) = (9,-3) w= (-3/2,1/2)

In each of Exercises 25-28, calculate the resultant force

F =Fi+ F2• What is the magnitude of F?

25. Fi= (3,0) 26. Fi= (1,2) 27. Fi=i-2j Fz= (0,4) Fz= (2,2) Fz = 2i + j 28. Fi=i-j Fz = -5i + 13j

In each of Exercises 29-32, determine the direction vector u that makes the given positive angle a with the positive x-axis.

29. 7r/6

30. 37r/4

3L 7r

32. 57r/3

In each of Exercises 33-36, write the given vector v in the form >.u where >. is a positive scalar, and u is a direction vector.

33. v = (6, -5/2)

34. v

=

(-Vil, 5) 35. v= 3i-2j 36. v=-

i-

v:.

j

In each of Exercises 37-40, write the given vector v in the form (a,b).

37. 2( 5i + 3j) - 3(i - j)

38. 4(i - 2j) + 5(i + 2j)

39. 3j - 2(-3i + j) -2i

40. 5j + 2(-j + 4i) -3(2j + i)

Further Theory and Practice

In Exercises 41-54, determine the value of a from the given information about v = (a, b).

4L a>b=3 and

ll

v

ll

=7

42. a < b = 5 and v = 13

43. v is parallel to (5, 3b) 44. v is parallel to (7, -b/2)

45. v is opposite in direction to w= (2, c) and

ll

v

ll

=3

ll

w

ll

46. v is opposite in direction to

=

(-3, c) and

ll

v

ll =

II II

47. both a and b are positive, and v is a diagonal of the parallelogram generated by (3, 5) and (-1, 7)

48. both a and b are negative, and v is a diagonal of the parallelogram generated by (4, 7) and (-1,8)

49. both a and b are positive, and v is a diagonal of the parallelogram generated by (-7, 6) and (3, 7)

50. both a and b are negative, and v is a diagonal of the parallelogram generated by (2, 9) and (-5, 3)

SL dir(v)

=

dir( )and

II

v

ii =

3 for w

=

(1, v'3)

52. dir(v)

=

dir(2w)and

ll

v

ll =

1/2 for w

=

(2, 1)

53. dir(v) = dir(-w) and

ll

v

ll

= 10 for w = (3, -4)

54. dir(v) = -dir(w) and

ll

v

ll

= 1/../2 for w = (-v'3, v's)

55. Mr. and Mrs. Woodman are pulling on ropes tied to a heavy wagon. Refer to Figure 14. If Mr. Woodman pulls with a strength of 100 pounds, then how hard must

(29)

Iman

A Figure 14

Mrs. Woodman pull so that the wagon moves along the dotted line? What is the magnitude of the resultant force?

56. Three forces

i,

F2, and F3 ·re applied to a point mass. Suppose that F1

=

lOOj newtons and tha1 r 2 has magni­

tude

120

newtons applied in the direction

5, 415

. What must F 3 be if the point mass

is

to remain at rest?

57. A boat maintains a straight course across a

500

m wide river at a rate of

50

m/min. The current pulls the boat down river at a rate of

20

m/min. When the boat docks on the other shore, how far down river will it have been carried?

58. A river

is

1

mile wide and flows south with a current of

3

miles per hour. What speed and heading should a motorboat adopt in order to cross the river in

10

minutes and reach a point on the opposite bank due east of its point of departure?

59. Bjarne, Leif, and Sammy are towing their vessel. The forces that they exert are directed along the tow lines, as indicated in Figure

15,

which also provides the magni­ tudes of their forces. (Note that the actual force vectors are not drawn in Figure

15.)

What is the resultant force?

A Figure 15

9.1 Vectors in the Plane 731 60. Let P

=

(P1.

pz)

and Q

= qi, qz)

be distinct points in the plane. Let M be the midpoint of the segment joining P

nd Q. Use the relationship

OM= OP+ iPQ

to deter­ mine the Cartesian coordinates of M.

61. Let

v

and

w

be two given nonzero vectors. Prove that there is a unique number such that

v +

>..w

is as short as possible. In other words, the function from JR o efined by H

v +

>.wll

ttains an absolute minimum value.

62. Let

m

and b be constants. Show that

v

is parallel to

PQ

for every pair of points

P

and Q on the line

y =

mx

+

b f

and only if

v =

>.., 1,

m)

for some scalar>...

63. Verify that

w

= - , ) is

perpendicular to

v =

a,

Describe all vectors that are perpendicular to

v.

64. Let

v

nd

w

be two nonzero vectors in the plane that are

not

parallel. Let

u

be any other vector. Prove that there

re unique scalars>.. andµ such that

u= .Av+ µw.

If

v =

i nd

w

=

j, then .A andµ are the entries of vector n.

In general, we can think of

v

and

w

generating a coordi­ nate system with }.. and µ, the entries of vector n in this

new coordinate system.)

65 and 66, develop a formula for the distance of a point to a line.

65. Suppose that

A

and

B

are not

0.

Consider the line l whose Cartesian equation

is Ax + By + D =

0.

Suppose that

Po= (.xo,y0)

does not lie on l. Show that n

= A, B

is a

vector that is perpendicular to l. Let Q"'

= x, -Ax/B­

DIB)

be a point on l. Calculate

Po�

�-For what value of

x is�

parallel to n?

66. For the value of� determined in Exercise

65,

show that

llP

-

o

'011

=Ax +By +D.

JA2 + B2

What does it mean to state that this is the distance of

Po

o l?

67. Let

P =(pi, pz)

and Q

= qi, qz)

be distinct points in the plane. Suppose that

0

1.

What are the Cartesian coordinates of the point

P,

on line segment PQ whose distance from Pis ·

PQ

? Use vectors to find an elegant

solution to this problem.

Let

A, B,

and

C

enote any three distinct points that are not collinear. Let a:, , and "I denote the midpoints of line

segments

BC, AC,

and

AB,

respectively. The line segments

Aa, B{3,

and

C"(

are called

medians

of

6.ABC.

Exercises 68-70 concern these medians and their common point of intersection, the

centroid

of

6.ABC.

4- - 4 ....+

68. Prove that

Ao: +

B/3

+ C =

0.

69. Let M e point of intersection of

Aa

ndB/3. Show that

AM=

2 3}.A

ndBM

=

2/3) nJJ.

Symmetrically, the same relationship holds when the medians

Bf3

nd

C"f

are

References

Related documents

Answer: We corrected the discussion taking into consideration comments of the reviewer.. I declare that there isn’t conflict of interest for the article: “Forensic botany as a

Quality: We measure quality (Q in our formal model) by observing the average number of citations received by a scientist for all the papers he or she published in a given

The purpose of this study was to find out the effect of using problem-based learning to enhance Junior High School students’ understanding and attitude toward

Witold Abramowicz, Pawel Kalczynski and Krzysztof We^cel. Filtering the

How the study was conducted The researchers used a 3-D global atmospheric download to predict how the radioactive material download move over earth and a health-effects model to see

This paper outlines the future possible scenarios that the air transport sector may evolve into after 2010, starting from the most important trends from the past and actual market

After testing all four filters as shown in Testing &amp; Construction, the wide bandwidth and decent phase accuracy of the 7 stage, 2.611 Notch-to-Notch ratio filter caused it to

Learning - helping people gain skills and competencies to drive performance Social collaboration - connecting employees to improve knowledge sharing and engagement.. Recruiting