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AdvancedDynamics

AdvancedDynamics

SchoolofAerospace,Mechanical,andManufacturingEngineering

RMITUniversity Melbourne,Australia

Thisbookisprintedonacid-freepaper.

Copyright c 2011byJohnWiley&Sons,Inc.Allrightsreserved.

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LibraryofCongressCataloging-in-PublicationData:

Jazar,RezaN. Advanceddynamics:rigidbody,multibody,andaerospaceapplications/RezaN.Jazar. p.cm. Includesindex.

ISBN978-0-470-39835-7(hardback);ISBN978-0-470-89211-4(ebk);ISBN978-0-470-89212-1(ebk); ISBN978-0-470-89213-8(ebk);ISBN978-0-470-95002-9(ebk);ISBN978-0-470-95159-0(ebk); ISBN978-0-470-95176-7(ebk) 1.Dynamics.I.Title. TA352.J392011 620.1 04—dc22

2010039778

PrintedintheUnitedStatesofAmerica

10987654321

Theansweriswaitingfortherightquestion.

Tomydaughter

Vazan , myson
Kavosh , andmywife, Mojgan

Prefacexiii

PartI Fundamentals1

1FundamentalsofKinematics3

1.1CoordinateFrameandPositionVector3

1.1.1Triad3

1.1.2CoordinateFrameandPositionVector4

1.1.3 VectorDefinition10

1.2VectorAlgebra12

1.2.1VectorAddition12

1.2.2VectorMultiplication17

1.2.3 IndexNotation26

1.3OrthogonalCoordinateFrames31

1.3.1OrthogonalityCondition31

1.3.2UnitVector34

1.3.3DirectionofUnitVectors36

1.4DifferentialGeometry37

1.4.1SpaceCurve38

1.4.2SurfaceandPlane43

1.5MotionPathKinematics46

1.5.1VectorFunctionandDerivative46

1.5.2VelocityandAcceleration51

1.5.3 NaturalCoordinateFrame54

1.6Fields77

1.6.1SurfaceandOrthogonalMesh78

1.6.2ScalarFieldandDerivative85

1.6.3VectorFieldandDerivative92 KeySymbols100 Exercises103

2FundamentalsofDynamics114

2.1LawsofMotion114

2.2EquationofMotion119

2.2.1ForceandMoment120

2.2.2MotionEquation125

Note:Astar( )indicatesamoreadvancedsubjectorexamplethatisnotdesignedforundergraduate teachingandcanbedroppedinthefirstreading.

2.3SpecialSolutions131

2.3.1ForceIsaFunctionofTime, F = F(t) 132

2.3.2ForceIsaFunctionofPosition, F = F(x) 141

2.3.3 EllipticFunctions148

2.3.4ForceIsaFunctionofVelocity, F = F(v) 156

2.4SpatialandTemporalIntegrals165

2.4.1SpatialIntegral:WorkandEnergy165

2.4.2TemporalIntegral:ImpulseandMomentum176

2.5 ApplicationofDynamics188

2.5.1 Modeling189

2.5.2 EquationsofMotion197

2.5.3 DynamicBehaviorandMethodsofSolution200

2.5.4 ParameterAdjustment220

KeySymbols223 Exercises226

PartII

GeometricKinematics241

3CoordinateSystems243

3.1CartesianCoordinateSystem243

3.2CylindricalCoordinateSystem250

3.3SphericalCoordinateSystem263

3.4 NonorthogonalCoordinateFrames269

3.4.1 ReciprocalBaseVectors269

3.4.2 ReciprocalCoordinateFrame278

3.4.3 InnerandOuterVectorProduct285

3.4.4 KinematicsinObliqueCoordinateFrames298

3.5 CurvilinearCoordinateSystem300

3.5.1 PrincipalandReciprocalBaseVectors301

3.5.2 Principal–ReciprocalTransformation311

3.5.3 CurvilinearGeometry320

3.5.4 CurvilinearKinematics325

3.5.5 KinematicsinCurvilinearCoordinates335 KeySymbols346 Exercises347

4RotationKinematics357

4.1RotationAboutGlobalCartesianAxes357

4.2SuccessiveRotationsAboutGlobalAxes363

4.3GlobalRoll–Pitch–YawAngles370

4.4RotationAboutLocalCartesianAxes373

4.5SuccessiveRotationsAboutLocalAxes376

4.6EulerAngles379

4.7LocalRoll–Pitch–YawAngles391

4.8LocalversusGlobalRotation395

4.9GeneralRotation397

4.10ActiveandPassiveRotations409

4.11 RotationofRotatedBody411

KeySymbols415

Exercises416

5OrientationKinematics422

5.1Axis–AngleRotation422

5.2EulerParameters438

5.3 Quaternion449

5.4 SpinorsandRotators457

5.5 ProblemsinRepresentingRotations459

5.5.1 RotationMatrix460

5.5.2 Axis–Angle461

5.5.3 EulerAngles462

5.5.4 QuaternionandEulerParameters463

5.6CompositionandDecompositionofRotations465

5.6.1CompositionofRotations466

5.6.2 DecompositionofRotations468

KeySymbols470

Exercises471

6MotionKinematics477

6.1Rigid-BodyMotion477

6.2HomogeneousTransformation481

6.3InverseandReverseHomogeneousTransformation494

6.4CompoundHomogeneousTransformation500

6.5 ScrewMotion517

6.6 InverseScrew529

6.7 CompoundScrewTransformation531

6.8 Pl ¨ uckerLineCoordinate534

6.9 GeometryofPlaneandLine540

6.9.1 Moment540

6.9.2 AngleandDistance541

6.9.3 PlaneandLine541

6.10 ScrewandPluckerCoordinate545

KeySymbols547

Exercises548

Contents

7MultibodyKinematics555

7.1MultibodyConnection555

7.2Denavit–HartenbergRule563

7.3ForwardKinematics584

7.4AssemblingKinematics615

7.5 Order-FreeRotation628

7.6 Order-FreeTransformation635

7.7 ForwardKinematicsbyScrew643

7.8 CasterTheoryinVehicles649

7.9InverseKinematics662

KeySymbols684 Exercises686

PartIII DerivativeKinematics693

8VelocityKinematics695

8.1AngularVelocity695

8.2TimeDerivativeandCoordinateFrames718

8.3MultibodyVelocity727

8.4VelocityTransformationMatrix739

8.5 DerivativeofaHomogeneousTransformationMatrix748

8.6 MultibodyVelocity754

8.7 Forward-VelocityKinematics757

8.8 Jacobian-GeneratingVector765

8.9 Inverse-VelocityKinematics778

KeySymbols782 Exercises783

9AccelerationKinematics788

9.1AngularAcceleration788

9.2SecondDerivativeandCoordinateFrames810

9.3MultibodyAcceleration823

9.4ParticleAcceleration830

9.5 MixedDoubleDerivative858

9.6 AccelerationTransformationMatrix864

9.7 Forward-AccelerationKinematics872

9.8 Inverse-AccelerationKinematics874

KeySymbols877 Exercises878

10Constraints887

10.1HomogeneityandIsotropy887

10.2DescribingSpace890

10.2.1ConfigurationSpace890

10.2.2EventSpace896

10.2.3StateSpace900

10.2.4State–TimeSpace908

10.2.5 KinematicSpaces910

10.3HolonomicConstraint913

10.4GeneralizedCoordinate923

10.5ConstraintForce932

10.6VirtualandActualWorks935

10.7 NonholonomicConstraint952

10.7.1 NonintegrableConstraint952

10.7.2 InequalityConstraint962

10.8 DifferentialConstraint966

10.9GeneralizedMechanics970

10.10 IntegralofMotion976

10.11 MethodsofDynamics996

10.11.1 LagrangeMethod996

10.11.2 GaussMethod999

10.11.3 HamiltonMethod1002

10.11.4 Gibbs–AppellMethod1009

10.11.5 KaneMethod1013

10.11.6 NielsenMethod1017

KeySymbols1021 Exercises1024

PartIV Dynamics1031

11RigidBodyandMassMoment1033

11.1RigidBody1033

11.2ElementsoftheMassMomentMatrix1035

11.3TransformationofMassMomentMatrix1044

11.4PrincipalMassMoments1058

KeySymbols1065 Exercises1066

12Rigid-BodyDynamics1072

12.1Rigid-BodyRotationalCartesianDynamics1072

12.2 Rigid-BodyRotationalEulerianDynamics1096

12.3Rigid-BodyTranslationalDynamics1101

12.4ClassicalProblemsofRigidBodies1112

12.4.1Torque-FreeMotion1112

12.4.2SphericalTorque-FreeRigidBody1115

12.4.3AxisymmetricTorque-FreeRigidBody1116

12.4.4 AsymmetricTorque-FreeRigidBody1128

12.4.5GeneralMotion1141

12.5MultibodyDynamics1157

12.6 RecursiveMultibodyDynamics1170

KeySymbols1177

Exercises1179

13LagrangeDynamics1189

13.1LagrangeFormofNewtonEquations1189

13.2LagrangeEquationandPotentialForce1203

13.3 VariationalDynamics1215

13.4 HamiltonPrinciple1228

13.5 LagrangeEquationandConstraints1232

13.6ConservationLaws1240

13.6.1ConservationofEnergy1241

13.6.2ConservationofMomentum1243

13.7 GeneralizedCoordinateSystem1244

13.8 MultibodyLagrangianDynamics1251

KeySymbols1262

Exercises1264

References1280

AGlobalFrameTripleRotation1287

BLocalFrameTripleRotation1289

CPrincipalCentralScrewTripleCombination1291

DIndustrialLinkDHMatrices1293

ETrigonometricFormula1300

Index1305

Preface

Thisbookisarrangedinsuchaway,andcoversthosematerials,thatIwouldhave likedtohavehadavailableasastudent:straightforward,righttothepoint,analyzing asubjectfromdifferentviewpoints,showingpracticalaspectsandapplicationofevery subject,consideringphysicalmeaningandsense,withinterestingandclearexamples. Thisbookwaswrittenforgraduatestudentswhowanttolearneveryaspectofdynamicsanditsapplication.Itisbasedontwodecadesofresearchandteachingcoursesin advanceddynamics,attitudedynamics,vehicledynamics,classicalmechanics,multibodydynamics,androbotics.

Iknowthatthebestwaytolearndynamicsisrepeatandpractice,repeatand practice.So,youaregoingtoseesomerepeatingandmuchpracticinginthisbook. Ibeginwithfundamentalsubjectsindynamicsandendwithadvancedmaterials.I introducethefundamentalknowledgeusedinparticleandrigid-bodydynamics.This knowledgecanbeusedtodevelopcomputerprogramsforanalyzingthekinematics, dynamics,andcontrolofdynamicsystems.

Thesubjectofrigidbodyhasbeenattheheartofdynamicssincethe1600s andremainsalivewithmoderndevelopmentsofapplications.Classicalkinematicsand dynamicshavetheirrootsintheworkofgreatscientistsofthepastfourcenturieswho establishedthemethodologyandunderstandingofthebehaviorofdynamicsystems. Thedevelopmentofdynamicscience,sincethebeginningofthetwentiethcentury,has movedtowardanalysisofcontrollableman-madeautonomoussystems.

LEVELOFTHEBOOK

Morethanhalfofthematerialisincommonwithcoursesinadvanceddynamics, classicalmechanics,multibodydynamics,andspacecraftdynamics.Graduatestudents inmechanicalandaerospaceengineeringhavethepotentialtoworkonprojectsthat arerelatedtoeitheroftheseengineeringdisciplines.However,studentshavenotseen enoughapplicationsinallareas.Althoughtheirtextbooksintroducerigid-bodydynamics,mechanicalengineeringstudentsonlyworkonengineeringapplicationswhile aerospaceengineeringstudentsonlyseespacecraftapplicationsandattitudedynamics.Thereaderofthistextwillhavenoprobleminanalyzingadynamicsysteminany oftheseareas.Thisbookbridgesthegapbetweenrigid-body,classical,multibody,and spacecraftdynamicsforgraduatestudentsandspecialistsinmechanicalandaerospace engineering.Engineersandgraduatestudentswhoreadthisbookwillbeabletoapply theirknowledgetoawiderangeofengineeringdisciplines.

Thisbookisaimedprimarilyatgraduatestudentsinengineering,physics,and mathematics.Itisespeciallyusefulforcoursesinthedynamicsofrigidbodiessuch asadvanceddynamics,classicalmechanics,attitudedynamics,spacecraftdynamics, andmultibodydynamics.Itprovidesbothfundamentalandadvancedtopicsonthe

kinematicsanddynamicsofrigidbodies.Thewholebookcanbecoveredintwo successivecourses;however,itispossibletojumpoversomesectionsandcoverthe bookinonecourse.

Thecontentsofthebookhavebeenkeptatafairlytheoretical–practicallevel. Manyconceptsaredeeplyexplainedandtheiruseemphasized,andmostoftherelated theoryandformalproofshavebeenexplained.Throughoutthebook,astrongemphasis isputonthephysicalmeaningoftheconceptsintroduced.Topicsthathavebeen selectedareofhighinterestinthefield.Anattempthasbeenmadetoexposethe studentstoabroadrangeoftopicsandapproaches.

ORGANIZATIONOFTHEBOOK

Thebookbeginswithareviewofcoordinatesystemsandparticledynamics.This introductionwillteachstudentstheimportanceofcoordinateframes.Transformation androtationtheoryalongwithdifferentiationtheoryindifferentcoordinateframeswill providetherequiredbackgroundtolearnrigid-bodydynamicsbasedonNewton–Euler principles.Themethodwillshowitsapplicationsinrigid-bodyandmultibodydynamics.TheNewtonequationsofmotionwillbetransformedtoLagrangianequationsas abridgetoanalyticaldynamics.ThemethodsofLagrangewillbeappliedonparticles andrigidbodies.

Throughitsexaminationofspecialistapplicationshighlightingthemanydifferent aspectsofdynamics,thistextprovidesanexcellentinsightintoadvancedsystems withoutrestrictingitselftoaparticulardiscipline.Theresultisessentialreadingfor allthoserequiringageneralunderstandingofthemoreadvancedaspectsofrigid-body dynamics.

Thetextisorganizedsuchthatitcanbeusedforteachingorforself-study.Part I“Fundamentals,”containsgeneralpreliminariesandprovidesadeepreviewofthe kinematicsanddynamics.AnewclassificationofvectorsisthehighlightofPartI.

PartII,“GeometricKinematics,”presentsthemathematicsofthedisplacementof rigidbodiesusingthematrixmethod.Theorder-freetransformationtheory,classificationofindustriallinks,kinematicsofsphericalwrists,andmechanicalsurgeryof multibodiesarethehighlightsofPartII.

PartIII,“DerivativeKinematics,”presentsthemathematicsofvelocityandaccelerationofrigidbodies.Thetimederivativesofvectorsindifferentcoordinateframes, Razıacceleration,integralsofmotion,andmethodsofdynamicsarethehighlightsof PartIII.

PartIV,“Dynamics,”presentsadetaileddiscussionofrigid-bodyandLagrangian dynamics.Rigid-bodydynamicsisstudiedfromdifferentviewpointstoprovidedifferentclassesofsolutions.Lagrangianmechanicsisreviewedindetailfromanapplied viewpoint.MultibodydynamicsandLagrangianmechanicsingeneralizedcoordinates arethehighlightsofPartIV.

METHODOFPRESENTATION

Thestructureofthepresentationisina fact–reason–application fashion.The“fact”is themainsubjectweintroduceineachsection.Thenthe“reason”isgivenasaproof.

Preface xv

Finallythe“application”ofthefactisexaminedinsomeexamples.Theexamplesare averyimportantpartofthebookbecausetheyshowhowtoimplementtheknowledge introducedinthefacts.Theyalsocoversomeothermaterialneededtoexpandthe subject.

PREREQUISITES

Thebookiswrittenforgraduatestudents,sotheassumptionisthatusersarefamiliar withthefundamentalsofkinematicsanddynamicsaswellasbasicknowledgeoflinear algebra,differentialequations,andthenumericalmethod.

UNITSYSTEM

Thesystemofunitsadoptedinthisbookis,unlessotherwisestated,theInternational SystemofUnits(SI).Theunitsofdegree(deg)andradian(rad)areutilizedforvariables representingangularquantities.

SYMBOLS

• Lowercaseboldlettersindicateavector.Vectorsmaybeexpressedinan ndimensionalEuclideanspace:

• Uppercaseboldlettersindicateadynamicvectororadynamicmatrix:

• Lowercaseletterswithahatindicateaunitvector.Unitvectorsarenotbolded:

• Lowercaseletterswithatildeindicatea3 × 3skewsymmetricmatrixassociated toavector:

• Anarrowabovetwouppercaselettersindicatesthestartandendpointsofa positionvector:

• Adoublearrowabovealowercaseletterindicatesa4 × 4matrixassociatedtoa quaternion:

• Thelengthofavectorisindicatedbyanonboldlowercaseletter:

• Capitalletters A, Q, R ,and T indicaterotationortransformationmatrices:

• Capitalletter B isutilizedtodenoteabodycoordinateframe: B(oxyz),B(Oxyz),B1 (o1 x1 y1 z1 )

• Capitalletter G isutilizedtodenoteaglobal,inertial,orfixedcoordinateframe:

G,G(XYZ),G(OXYZ)

• Rightsubscriptonatransformationmatrixindicatesthe departure frames: TB = transformationmatrixfromframe B(oxyz)

• Leftsuperscriptonatransformationmatrixindicatesthe destination frame: G TB = transformationmatrixfromframe B(oxyz) toframe G(OXYZ)

• Wheneverthereisnosubscriptorsuperscript,thematricesareshowninbrackets:

• Leftsuperscriptonavectordenotestheframeinwhichthevectorisexpressed. Thatsuperscriptindicatestheframethatthevectorbelongsto,sothevectoris expressedusingtheunitvectorsofthatframe:

G r = positionvectorexpressedinframe G(OXYZ)

• Rightsubscriptonavectordenotesthetippointtowhichthevectorisreferred:

G rP = positionvectorofpoint P expressedincoordinateframe G(OXYZ)

• Rightsubscriptonanangularvelocityvectorindicatestheframetowhichthe angularvectorisreferred:

ωB = angularvelocityofthebodycoordinateframe B(oxyz)

• Leftsubscriptonanangularvelocityvectorindicatestheframewithrespectto whichtheangularvectorismeasured:

G ωB = angularvelocityofthebodycoordinateframe B(oxyz) withrespecttotheglobalcoordinateframe G(OXYZ)

• Leftsuperscriptonanangularvelocityvectordenotestheframeinwhichthe angularvelocityisexpressed:

B2 G ωB1 = angularvelocityofthebodycoordinateframe B1 withrespecttotheglobalcoordinateframe G andexpressedinbodycoordinateframe B2

Wheneverthesubscriptandsuperscriptofanangularvelocityarethesame,we usuallydroptheleftsuperscript:

ωB ≡ G G ωB

Alsoforposition,velocity,andaccelerationvectors,wedroptheleftsubscripts ifitisthesameastheleftsuperscript:

• Iftherightsubscriptonaforcevectorisanumber,itindicatesthenumberof coordinateframesinaserialrobot.Coordinateframe Bi issetupatjoint i + 1:

Fi = forcevectoratjoint i + 1measuredattheoriginof Bi (oxyz)

Atjoint i thereisalwaysanactionforce Fi thatlink(i )appliesonlink(i + 1) andareactionforce Fi thatlink(i + 1)appliesonlink(i ).Onlink(i )thereis alwaysanactionforce Fi 1 comingfromlink(i 1)andareactionforce Fi comingfromlink(i + 1).Theactionforceiscalledthe drivingforce ,andthe reactionforceiscalledthe drivenforce

• Iftherightsubscriptonamomentvectorisanumber,itindicatesthenumberof coordinateframesinaserialrobot.Coordinateframe Bi issetupatjoint i + 1:

Mi = momentvectoratjoint i + 1measuredattheoriginof Bi (oxyz)

Atjoint i thereisalwaysanactionmoment Mi thatlink(i )appliesonlink (i + 1),andareactionmoment Mi thatlink(i + 1)appliesonlink(i ).On

G
B B vP ≡ B vP

link(i )thereisalwaysanactionmoment Mi 1 comingfromlink(i 1)anda reactionmoment Mi comingfromlink(i + 1).Theactionmomentiscalledthe drivingmoment ,andthereactionmomentiscalledthe drivenmoment

• Leftsuperscriptonderivativeoperatorsindicatestheframeinwhichthederivative ofavariableistaken:

Ifthevariableisavectorfunctionandtheframeinwhichthevectorisdefinedis thesameastheframeinwhichatimederivativeistaken,wemayusetheshort notation

andwriteequationssimpler.Forexample,

• Iffollowedbyangles,lowercase c and s denotecosandsinfunctionsinmathematicalequations:

• Capitalboldletter I indicatesaunitmatrix,which,dependingonthedimension ofthematrixequation,couldbea3 × 3ora4 × 4unitmatrix. I3 or I4 arealso beingusedtoclarifythedimensionof I.Forexample,

• Twoparalleljointaxesareindicatedbyaparallelsign( ).

• Twoorthogonaljointaxesareindicatedbyanorthogonalsign( ).Twoorthogonal jointaxesareintersectingatarightangle.

• Twoperpendicularjointaxesareindicatedbyaperpendicularsign(⊥).Two perpendicularjointaxesareatarightanglewithrespecttotheircommonnormal.

PartI

Fundamentals

Therequiredfundamentalsofkinematicsanddynamicsarereviewedinthispart. Itshouldprepareusforthemoreadvancedparts.

FundamentalsofKinematics

Vectorsandcoordinateframesarehuman-madetoolstostudythemotionofparticles andrigidbodies.Weintroducetheminthischaptertoreviewthefundamentalsof kinematics.

1.1COORDINATEFRAMEANDPOSITIONVECTOR

Toindicatethepositionofapoint P relativetoanotherpoint O inathree-dimensional (3D)space,weneedtoestablishacoordinateframeandprovidethreerelativecoordinates.Thethreecoordinatesarescalarfunctionsandcanbeusedtodefineaposition vectorandderiveotherkinematiccharacteristics.

1.1.1Triad

Takefournon-coplanarpoints O , A, B , C andmakethreelines OA, OB , OC .The triadOABC isdefinedbytakingthelines OA, OB , OC asarigidbody.Theposition of A isarbitraryprovideditstaysonthesamesideof O .Thepositionsof B and C are similarlyselected.Nowrotate OB about O intheplane OAB sothattheangle AOB becomes90deg.Next,rotate OC aboutthelinein AOB towhichitisperpendicular untilitbecomesperpendiculartotheplane AOB .Thenewtriad OABC iscalledan orthogonaltriad .

Havinganorthogonaltriad OABC ,anothertriad OA BC maybederivedbymoving A totheothersideof O tomakethe oppositetriadOA BC .Allorthogonaltriadscan besuperposedeitheronthetriad OABC oronitsopposite OA BC .

Oneofthetwotriads OABC and OA BC canbedefinedasbeinga positivetriad andusedasa standard .Theotheristhendefinedasa negativetriad .Itisimmaterial whichoneischosenaspositive;however,usuallythe right-handedconvention ischosen aspositive.Theright-handedconventionstatesthatthedirectionofrotationfrom OA to OB propelsa right-handedscrew inthedirection OC .Aright-handedorpositive orthogonaltriadcannotbesuperposedtoaleft-handedornegativetriad.Therefore, thereareonlytwoessentiallydistincttypesoftriad.Thisisapropertyof3Dspace.

Weuseanorthogonaltriad OABC withscaledlines OA, OB , OC tolocateapoint in3Dspace.Whenthethreelines OA, OB , OC havescales,thensuchatriadiscalled a coordinateframe .

Everymovingbodyiscarryinga moving or bodyframe thatisattachedtothebody andmoveswiththebody.Abodyframeacceptseverymotionofthebodyandmay alsobecalleda localframe .Thepositionandorientationofabodywithrespectto otherframesisexpressedbythepositionandorientationofitslocalcoordinateframe.

Whenthereareseveralrelativelymovingcoordinateframes,wechooseoneofthem asa referenceframe inwhichweexpressmotionsandmeasurekinematicinformation. Themotionofabodymaybeobservedandmeasuredindifferentreferenceframes; however,weusuallycomparethemotionofdifferentbodiesinthe globalreference frame .Aglobalreferenceframeisassumedtobemotionlessandattachedtotheground.

Example1CyclicInterchangeofLetters Inanyorthogonaltriad OABC ,cyclic interchangingoftheletters ABC produceanotherorthogonaltriadsuperposableonthe originaltriad.Cyclicinterchangingmeansrelabeling A as B , B as C ,and C as A or pickinganythreeconsecutivelettersfrom ABCABCABC ....

Example2 IndependentOrthogonalCoordinateFrames

Havingonlytwotypes oforthogonaltriadsin3Dspaceisassociatedwiththefactthataplanehasjusttwo sides.Inotherwords,therearetwooppositenormaldirectionstoaplane.Thismay alsobeinterpretedas:wemayarrangetheletters A, B ,and C injusttwoorderswhen cyclicinterchangeisallowed:

ABC , ACB

Ina4Dspace,therearesixcyclicordersforfourletters A,B,C ,and D :

ABCD , ABDC , ACBD , ACDB , ADBC , ADCB

So,therearesixdifferenttetradsina4Dspace.

Inan n Dspacethereare (n 1)!cyclicordersfor n letters,sothereare (n 1)! differentcoordinateframesinan n Dspace.

Example3Right-HandRule

Aright-handedtriadcanbeidentifiedbyaright-hand rulethatstates:Whenweindicatethe OC axisofanorthogonaltriadbythethumbof therighthand,theotherfingersshouldturnfrom OA to OB tocloseourfist.

Theright-handrulealsoshowstherotationofEarthwhenthethumboftheright handindicatesthenorthpole.

Pushyourrightthumbtothecenterofaclock,thentheotherfingerssimulatethe rotationoftheclock’shands.

Pointyourindexfingeroftherighthandinthedirectionofanelectriccurrent. Thenpointyourmiddlefingerinthedirectionofthemagneticfield.Yourthumbnow pointsinthedirectionofthemagneticforce.

Ifthethumb,indexfinger,andmiddlefingeroftherighthandareheldsothat theyformthreerightangles,thenthethumbindicatesthe Z -axiswhentheindexfinger indicatesthe X -axisandthemiddlefingerthe Y -axis.

1.1.2CoordinateFrameandPositionVector

Considerapositiveorthogonaltriad OABC asisshowninFigure1.1.Weselecta unit length anddefinea directedline ˆ ı on OA withaunitlength.Apoint P1 on OA isat adistance x from O suchthatthedirectedline →OP 1 from O to P1 is →OP 1 = x ˆ ı .The

Figure1.1 Apositiveorthogonaltriad OABC ,unitvectors ˆ ı , ˆ  , ˆ k ,andapositionvector r with components x , y , z .

directedline ˆ ı iscalleda unitvector on OA,theunitlengthiscalledthe scale ,point O iscalledthe origin ,andtherealnumber x iscalledthe ˆ ı -coordinate of P1 .The distance x mayalsobecalledthe ˆ ı measurenumber of →OP 1 .Similarly,wedefinethe unitvectors ˆ  and ˆ k on OB and OC anduse y and z astheircoordinates,respectively. Althoughitisnotnecessary,weusuallyusethesamescalefor ˆ ı , ˆ  , ˆ k andreferto OA, OB , OC by ˆ ı , ˆ  , ˆ k andalsoby x , y , z .

Thescalarcoordinates x , y , z arerespectivelythelengthofprojectionsof P on OA, OB ,and OC andmaybecalledthe components of r.Thecomponents x , y , z are independentandwemayvaryanyofthemwhilekeepingtheothersunchanged.

Ascaledpositiveorthogonaltriadwithunitvectors ˆ ı , ˆ  , ˆ k iscalledan orthogonal coordinateframe .Thepositionofapoint P withrespectto O isdefinedbythree coordinates x , y , z andisshownbya positionvector r = rP :

Toworkwithmultiplecoordinateframes,weindicatecoordinateframesbyacapital letter,suchas G and B ,toclarifythecoordinateframeinwhichthevector r is expressed.Weshowthenameoftheframeasaleftsuperscripttothevector:

Avector r isexpressedinacoordinateframe B onlyifitsunitvectors ˆ ı , ˆ  , ˆ k belong totheaxesof B .Ifnecessary,weusealeftsuperscript B andshowtheunitvectors as B

toindicatethat

Wemaydropthesuperscript B aslongaswehavejustonecoordinateframe. Thedistancebetween O and P isascalarnumber r thatiscalledthe length , magnitude , modulus , norm ,or absolutevalue ofthevector r:

Wemaydefineanewunitvector ˆ ur on r andshow r by r = r ˆ ur

Theequation r = r ˆ ur iscalledthe naturalexpression of r,whiletheequation r = x ˆ ı + y ˆ  + z ˆ k iscalledthe decomposition or decomposedexpression of r overtheaxes

ˆ ı , ˆ  , ˆ k .Equating(1.1)and(1.5)showsthat

Becausethelengthof ˆ ur isunity,thecomponentsof ˆ ur arethecosinesoftheangles

3 between

and

,respectively:

Thecosinesoftheangles α1 , α2 , α3 arecalledthe directionalcosines of ˆ ur ,which,as isshowninFigure1.1,arethesameasthedirectionalcosinesofanyothervectoron thesameaxisas ˆ ur ,including r.

Equations(1.7)–(1.9)indicatethatthethreedirectionalcosinesarerelatedbythe equation

Example4PositionVectorofaPointP Considerapoint P withcoordinates x = 3, y = 2, z = 4.Thepositionvectorof P is

Thedistancebetween O and P is

andtheunitvector ˆ ur on r is

Thedirectionalcosinesof ˆ ur are cos α1 = x r = 0.55708 cos α2 = y r = 0 37139(1.14) cos α3 = z r = 0.74278

andthereforetheanglesbetween r andthe x -, y -, z -axesare

Example5DeterminationofPosition Figure1.2illustratesapoint P inascaled triad OABC .Wedeterminethepositionofthepoint P withrespectto O by:

1. Drawingaline PD parallel OC tomeettheplane AOB at D

2. Drawing DP 1 parallelto OB tomeet OA at P1

Figure1.2 Determinationofposition.

Thelengths OP 1 , P1 D , DP arethecoordinatesof P anddetermineitspositionin triad OABC .Thelinesegment OP isadiagonalofaparallelepipedwith OP 1 , P1 D , DP asthreeedges.Thepositionof P isthereforedeterminedbymeansofaparallelepiped whoseedgesareparalleltothelegsofthetriadandoneofitsdiagonalistheline joiningtheorigintothepoint.

Example6VectorsinDifferentCoordinateFrames Figure1.3illustratesaglobally fixedcoordinateframe G atthecenterofarotatingdisc O .Anothersmallerrotating discwithacoordinateframe B isattachedtothefirstdiscataposition GdO .Point P isontheperipheryofthesmalldisc.

Figure1.3 Agloballyfixedframe G atthecenterofarotatingdisc O andacoordinateframe B atthecenterofamovingdisc.

Ifthecoordinateframe G(OXYZ ) isfixedand B(oxyz ) isalwaysparallelto G , thepositionvectorsof P indifferentcoordinateframesareexpressedby

maybeindicatedbyapositionvector

Example7VariableVectors Therearetwowaysthatavectorcanvary:lengthand direction.Avariable-lengthvectorisavectorinthenaturalexpressionwhereitsmagnitudeisvariable,suchas

Theaxisofavariable-lengthvectorisfixed.

Avariable-directionvectorisavectorinitsnaturalexpressionwheretheaxisofits unitvectorvaries.Toshowsuchavariablevector,weusethedecomposedexpression oftheunitvectorandshowthatitsdirectionalcosinesarevariable:

Theaxisanddirectioncharacteristicsarenotfixedforavariable-directionvector,while itsmagnituderemainsconstant.Theendpointofavariable-directionvectorslideson aspherewithacenteratthestartingpoint.

Avariablevectormayhaveboththelengthanddirectionvariables.Suchavector isshowninitsdecomposedexpressionwithvariablecomponents:

Itcanalsobeshowninitsnaturalexpressionwithvariablelengthanddirection:

Example8ParallelandPerpendicularDecompositionofaVector Consideraline l andavector r intersectingattheoriginofacoordinateframesuchasshownisin

Figure1.4.Theline l andvector r indicateaplane (l, r).Wedefinetheunitvectors

ˆ u parallelto l and ˆ u⊥ perpendicularto l inthe (l, r)-plane.Iftheanglebetween r and l is α ,thenthecomponentof r parallelto l is

andthecomponentof r perpendicularto l is

Thesecomponentsindicatethatwecandecomposeavector r toitsparallelandperpendicularcomponentswithrespecttoaline l byintroducingtheparallelandperpendicular unitvectors ˆ u and ˆ u⊥ :

Figure1.4 Decompositionofavector r withrespecttoaline l intoparallelandperpendicular components.

1.1.3 VectorDefinition

Byavectorwemeananyphysicalquantitythatcanberepresentedbyadirectedsection ofalinewithastartpoint,suchas O ,andanendpoint,suchas P .Wemayshowa vectorbyanorderedpairofpointswithanarrow,suchas → OP .Thesign → PP indicates azerovectoratpoint P . Lengthanddirectionarenecessarytohaveavector;however,avectormayhave fivecharacteristics:

1. Length .Thelengthofsection OP correspondstothemagnitudeofthephysical quantitythatthevectorisrepresenting.

2. Axis .Astraightlinethatindicatesthelineonwhichthevectoris.Thevector axisisalsocalledthe lineofaction .

3. Endpoint .Astartoranendpointindicatesthepointatwhichthevectoris applied.Suchapointiscalledthe affectingpoint .

4. Direction .Thedirectionindicatesatwhatdirectionontheaxisthevectoris pointing.

5. Physicalquantity .Anyvectorrepresentsaphysicalquantity.Ifaphysicalquantitycanberepresentedbyavector,itiscalleda vectorialphysicalquantity Thevalueofthequantityisproportionaltothelengthofthevector.Having avectorthatrepresentsnophysicalquantityismeaningless,althoughavector maybedimensionless.

Dependingonthephysicalquantityandapplication,thereareseventypesof vectors:

1. Vecpoint .Whenallofthevectorcharacteristics—length,axis,endpoint,direction,andphysicalquantity—arespecified,thevectoriscalleda boundedvector , pointvector ,or vecpoint .Suchavectorisfixedatapointwithnomovability.

2. Vecline .Ifthestartandendpointsofavectorarenotfixedonthevectoraxis, thevectoriscalleda slidingvector , linevector ,or vecline .Aslidingvectoris freetoslideonitsaxis.

3. Vecface .Whentheaffectingpointofavectorcanmoveonasurfacewhile thevectordisplacesparalleltoitself,thevectoriscalleda surfacevector or vecface .Ifthesurfaceisaplane,thenthevectorisa planevector or veclane

4. Vecfree .Iftheaxisofavectorisnotfixed,thevectoriscalleda freevector , directionvector ,or vecfree .Suchavectorcanmovetoanypointofaspecified spacewhileitremainsparalleltoitselfandkeepsitsdirection.

5. Vecpoline .Ifthestartpointofavectorisfixedwhiletheendpointcanslide onaline,thevectorisa point-linevector or vecpoline .Suchavectorhasa constraintvariablelengthandorientation.However,ifthestartandendpoints ofavecpolineareontheslidingline,itsorientationisconstant.

6. Vecpoface .Ifthestartpointofavectorisfixedwhiletheendpointcanslide onasurface,thevectorisa point-surfacevector or vecpoface .Suchavector hasaconstraintvariablelengthandorientation.Thestartandendpointsofa vecpofacemaybothbeontheslidingsurface.Ifthesurfaceisaplane,the vectoriscalleda point-planevector or vecpolane .

7. Vecporee .Whenthestartpointofavectorisfixedandtheendpointcan moveanywhereinaspecifiedspace,thevectoriscalleda point-freevector or vecporee .Suchavectorhasavariablelengthandorientation.

Figure1.5illustratesavecpoint,avecline,vecface,andavecfreeandFigure1.6 illustratesavecpoline,avecpoface,andavecporee.

Wemaycomparetwovectorsonlyiftheyrepresentthesamephysicalquantityand areexpressedinthesamecoordinateframe.Twovectorsareequaliftheyarecomparableandarethesametypeandhavethesamecharacteristics.Twovectorsareequivalent iftheyarecomparableandthesametypeandcanbesubstitutedwitheachother.

Insummary,anyphysicalquantitythatcanberepresentedbyadirectedsection ofalinewithastartandanendpointisavectorquantity.Avectormayhavefive characteristics:length,axis,endpoint,direction,andphysicalquantity.Thelengthand directionarenecessary.Thereareseventypesofvectors:vecpoint,vecline,vecface, vecfree,vecpoline,vecpoface,andvecporee.Vectorscanbeaddedwhentheyare coaxial.Incasethevectorsarenotcoaxial,thedecomposedexpressionofvectors mustbeusedtoaddthevectors.

Example9ExamplesofVectorTypes Displacement isavecpoint.Movingfroma point A toapoint B iscalledthedisplacement.Displacementisequaltothedifference oftwopositionvectors.A positionvector startsfromtheoriginofacoordinateframe

Figure1.5 (a )Avecpoint,(b )avecline,(c )avecface,and(d )avecfree.
Figure1.6 (a )avecpoline,(b )vecpoface,(c )vecporee.

andendsasapointintheframe.Ifpoint A isat rA andpoint B at rB ,thendisplacement from A to B is

Forceisavecline.InNewtonianmechanics,aforcecanbeappliedonabodyat anypointofitsaxisandprovidesthesamemotion.

Torqueisanexampleofvecfree.InNewtonianmechanics,amomentcanbeapplied onabodyatanypointparalleltoitselfandprovidesthesamemotion.

Aspacecurveisexpressedbyavecpoline,asurfaceisexpressedbyavecpoface, andafieldisexpressedbyavecporee.

Example10Scalars

Physicalquantitieswhichcanbespecifiedbyonlyanumber arecalled scalars .Ifaphysicalquantitycanberepresentedbyascalar,itiscalled a scalaricphysicalquantity .Wemaycomparetwoscalarsonlyiftheyrepresentthe samephysicalquantity.Temperature,density,andworkaresomeexamplesofscalaric physicalquantities.

Twoscalarsareequaliftheyrepresentthesamescalaricphysicalquantityandthey havethesamenumberinthesamesystemofunits.Twoscalarsareequivalentifwe cansubstituteonewiththeother.Scalarsmustbeequaltobeequivalent.

1.2VECTORALGEBRA

Mostofthephysicalquantitiesindynamicscanberepresentedbyvectors.Vectoraddition,multiplication,anddifferentiationareessentialforthedevelopmentofdynamics. Wecancombinevectorsonlyiftheyarerepresentingthesamephysicalquantity,they arethesametype,andtheyareexpressedinthesamecoordinateframe.

1.2.1VectorAddition

Twovectorscanbe added whentheyare coaxial .Theresultisanothervectoronthe sameaxiswithacomponentequaltothesumofthecomponentsofthetwovectors. Considertwocoaxialvectors r1 and r2 innaturalexpressions:

Theiradditionwouldbeanewvector

Because r1 and r2 arescalars,wehave r

,andtherefore,coaxialvector additionis commutative ,

andalso associative ,

Whentwovectors r1 and r2 arenotcoaxial,weusetheirdecomposedexpressions

andaddthecoaxialvectors

towritetheresult asthedecomposedexpressionof r3 = r1 + r2 :

So,thesumoftwovectors r1 and r2 isdefinedasavector r3 whereitscomponents areequaltothesumoftheassociatedcomponentsof r1 and r2 .Figure1.7illustrates vectoraddition r3 = r1 + r2 oftwovecpoints r1 and r2

Subtractionoftwovectorsconsistsofaddingtotheminuendthesubtrahendwith theoppositesense:

Thevectors r2 and r2 havethesameaxisandlengthanddifferonlyinhavingopposite direction.

Ifthecoordinateframeisknown,thedecomposedexpressionofvectorsmayalso beshownbycolumnmatricestosimplifycalculations:

Figure1.7 Vectoradditionoftwovecpoints r1 and r2 .

Vectorscanbeaddedonlywhentheyareexpressedinthesameframe.Thus,a vectorequationsuchas

ismeaninglesswithoutindicatingthatallofthemareexpressedinthesameframe, suchthat

Thethreevectors r1 , r2 ,and r3 arecoplanar,and r3 maybeconsideredasthe diagonalofaparallelogramthatismadeby

Example11DisplacementofaPoint Point P movesfromtheoriginofaglobal coordinateframe G toapointat (1, 2, 0) andthenmovesto (4, 3, 0).Ifweexpressthe firstdisplacementbyavector r1 anditsfinalpositionby r3 ,theseconddisplacement is r2 ,where

Example12VectorInterpolationProblem Havingtwodigits n1 and n2 asthestart andthefinalinterpolants,wemaydefineacontrolleddigit n withavariable q suchthat

Definingordeterminingsuchacontrolleddigitiscalledtheinterpolationproblem. Therearemanyfunctionstobeusedforsolvingtheinterpolationproblem.Linear interpolationisthesimplestandiswidelyusedinengineeringdesign,computer graphics,numericalanalysis,andoptimization:

Thecontrolparameter q determinestheweightofeachinterpolants n1 and n2 inthe interpolated n .Inalinearinterpolation,theweightfactorsareproportionaltothe distanceof q from1and0.

Figure1.8 Vectorlinearinterpolation.

Employingthelinearinterpolationtechnique,wemaydefineavector r = r (q ) to interpolatebetweentheinterpolantvectors r1 and r2 :

Inthisinterpolation,weassumedthatequalstepsin q resultsinequalstepsin r between r1 and r2 .Thetippointof r willmoveonalineconnectingthetippointsof r1 and r2 ,asisshowninFigure1.8.

Wemayinterpolatethevectors r1 and r2 byinterpolatingtheangulardistance θ between r1 and r2 :

ToderiveEquation(1.44),wemaystartwith

andfind a and b fromthefollowingtrigonometricequations:

Example13VectorAdditionandLinearSpace Vectorsandaddingoperationmake a linearspace becauseforanyvectors r1 , r2 wehavethefollowingproperties:

1. Commutative:

2. Associative:

3. Nullelement: 0 + r = r (1.50)

4. Inverseelement: r + ( r) = 0 (1.51)

Example14LinearDependenceandIndependence The n vectors r1 , r2 , r3 ,..., rn are linearlydependent ifthereexist n scalars c1 ,c2 ,c3 ,...,cn notallequaltozero suchthatalinearcombinationofthevectorsequalszero:

0(1.52)

Thevectors r1 , r2 , r3 ,..., rn are linearlyindependent iftheyarenotlinearlydependent, anditmeansthe n scalars c1 ,c2 ,c3 ,...,cn mustallbezerotohaveEquation(1.52):

Example15TwoLinearlyDependentVectorsAreColinear Considertwolinearly dependentvectors r1 and r2 :

+

= 0(1.54) If c1 = 0,wehave

(1.55) andif c2 = 0,wehave

whichshows r1 and r2 arecolinear.

Example16ThreeLinearlyDependentVectorsAreCoplanar Considerthreelinearly dependentvectors r1 , r2 ,and r3 ,

whereatleastoneofthescalars c1 ,c2 ,c3 ,say c3 ,isnotzero;then

whichshows r3 isinthesameplaneas r1 and r2 .

1.2.2VectorMultiplication

Therearethreetypesofvectormultiplicationsfortwovectors r1 and r2 :

1. Dot,Inner,orScalarProduct

Theinnerproductoftwovectorsproducesascalarthatisequaltotheproduct ofthelengthofindividualvectorsandthecosineoftheanglebetweenthem. Thevectorinnerproductis commutative inorthogonalcoordinateframes,

Theinnerproductisdimensionfreeandcanbecalculatedin n -dimensional spaces.Theinnerproductcanalsobeperformedinnonorthogonalcoordinate systems.

2. Cross,Outer,orVectorProduct

Theouterproductoftwovectors r1 and r2 producesanothervector r3 that isperpendiculartotheplaneof r1 , r2 suchthatthecycle r1 r2 r3 makesa right-handedtriad.Thelengthof r3 isequaltotheproductofthelengthof individualvectorsmultipliedbythesineoftheanglebetweenthem.Hence r3 isnumericallyequaltotheareaoftheparallelogrammadeupof r1 and r2 . Thevectorinnerproductis skewcommutative or anticommutative :

Theouterproductisdefinedandappliedonlyin3Dspace.Thereisno outerproductinlowerorhigherdimensionsthan3.Ifanyvectorof r1 and r2 isinalowerdimensionthan3D,wemustmakeita3Dvectorbyaddingzero componentsformissingdimensionstobeabletoperformtheirouterproduct.

3. QuaternionProduct

WewilltalkaboutthequaternionproductinSection5.3.

Insummary,therearethreetypesofvectormultiplication:inner,outer,andquaternionproducts,ofwhichtheinnerproductistheonlyonewithcommutativeproperty.

Example17GeometricExpressionofInnerProducts Consideraline l andavector r intersectingattheoriginofacoordinateframeasisshowninFigure1.9.Iftheangle between r and l is α ,theparallelcomponentof r to l is

Thisisthelengthoftheprojectionof r on l .Ifwedefineaunitvector ˆ ul on l byits directioncosines

thentheinnerproductof r and ˆ ul is

Wemayshow r byusingitsdirectioncosines

3 ,

Then,wemayusetheresultoftheinnerproductof r and ˆ ul ,

tocalculatetheangle α between r and l basedontheirdirectionalcosines:

Figure1.9 Aline l andavector r intersectingattheoriginofacoordinateframe.

So,theinnerproductcanbeusedtofindtheprojectionofavectoronagivenline.It isalsopossibletousetheinnerproducttodeterminetheangle α betweentwogiven vectors r1 and r2 as

Example18Power2ofaVector Bywritingavector r toapower2,wemeanthe innerproductof r toitself:

Usingthisdefinitionwecanwrite

Thereisnomeaningforavectorwithanegativeorpositiveoddexponent.

Example19UnitVectorsandInnerandOuterProducts Usingthesetofunitvectors

ˆ ı , ˆ  , ˆ k ofapositiveorthogonaltriadandthedefinitionofinnerproduct,weconcludethat

Furthermore,bydefinitionofthevectorproductwehave

Itmightalsobeusefulifwehavetheseequalities:

Example20VanishingDotProduct Iftheinnerproductoftwovectors a and b iszero, a · b = 0(1.81)

theneither a = 0or b = 0,or a and b areperpendicular.

Example21VectorEquations Assume x isanunknownvector, k isascalar,and a, b,and c arethreeconstantvectorsinthefollowingvectorequation:

Tosolvetheequationfor x,wedotproductbothsidesof(1.82)by b:

Thisisalinearequationfor x · b withthesolution

provided

Substituting(1.84)in(1.82)providesthesolution x:

Analternativemethodisdecompositionofthevectorequationalongtheaxes

, ˆ k ofthecoordinateframeandsolvingasetofthreescalarequationstofindthe componentsoftheunknownvector.

Assumethedecomposedexpressionofthevectors x, a, b,and c are

SubstitutingtheseexpressionsinEquation(1.82),

providesasetofthreescalarequations

thatcanbesolvedbymatrixinversion:

Solution(1.90)iscompatiblewithsolution(1.86).

Example22VectorAddition,ScalarMultiplication,andLinearSpace Vectoradditionandscalarmultiplicationmakealinearspace,because

Example23VanishingConditionofaVectorInnerProduct Considerthreenoncoplanarconstantvectors a, b, c andanarbitraryvector r.If

then

Example24VectorProductExpansion Wemayprovetheresultoftheinnerand outerproductsoftwovectorsbyusingdecomposedexpressionandexpansion:

Wemayalsofindtheouterproductoftwovectorsbyexpandingadeterminantand derivethesameresultasEquation(1.101):

Example25bac–cabRule If a, b, c arethreevectors,wemayexpandtheirtriple crossproductandshowthat

because

Equation(1.103)maybereferredtoasthe bac–cabrule ,whichmakesiteasyto remember.Thebac–cabruleisthemostimportantin3Dvectoralgebra.Itisthekey toproveagreatnumberofothertheorems.

Example26GeometricExpressionofOuterProducts Considerthefreevectors r1 from A to B and r2 from A to C ,asareshowninFigure1.10:

Thecrossproductofthetwovectorsis r3 :

where r3 = 8.4558isnumericallyequivalenttothearea A oftheparallelogram ABCD madebythesides AB and AC :

Theareaofthetriangle ABC is A/2.Thevector r3 isperpendiculartothisplaneand, hence,itsunitvector ˆ ur3 canbeusedtoindicatetheplane ABCD .

Example27ScalarTripleProduct Thedotproductofavector r1 withthecross productoftwovectors r2 and r3 iscalledthe scalartripleproduct of r1 , r2 ,and r3 . Thescalartripleproductcanbeshownandcalculatedbyadeterminant:

Interchangingtworows(orcolumns)ofamatrixchangesthesignofitsdeterminant. So,wemayconcludethatthescalartripleproductofthreevectors r1 , r2 , r3 isalso equalto

BecauseofEquation(1.111),thescalartripleproductofthevectors r1 , r2 , r3 canbe shownbytheshortnotation [

]:

Thisnotationgivesusthefreedomtosetthepositionofthedotandcrossproductsigns asrequired.

Ifthethreevectors r1 , r2 , r3 arepositionvectors,thentheirscalartripleproduct geometricallyrepresentsthevolumeoftheparallelepipedformedbythethreevectors. Figure1.11illustratessuchaparallelepipedforthreevectors r1 , r2 , r3 .

Figure1.11 Theparallelepipedmadebythreevectors r1 , r2 , r3 .

Example28VectorTripleProduct Thecrossproductofavector r1 withthecross productoftwovectors r2 and r3 iscalledthe vectortripleproduct of r1 , r2 ,and r3 . The bac–cab ruleisalwaysusedtosimplifyavectortripleproduct:

Example29 NormandVectorSpace Assume r, r1 , r2 , r3 arearbitraryvectors and c , c1 , c3 arescalars.The norm ofavector r isdefinedasareal-valuedfunction onavectorspace v suchthatforall {r1 , r2 } ∈ V andall c ∈ R wehave:

1. Positivedefinition: r > 0if r = 0and r = 0if r = 0.

2. Homogeneity: c r = c r

3. Triangleinequality: r1 + r2 = r1 + r2

Thedefinitionofnormisuptotheinvestigatorandmayvarydependingonthe application.Themostcommondefinitionofthenormofavectoristhelength: r = |r| = r 2 1 + r 2 2 + r 2 3 (1.114)

Theset v withvectorelementsiscalleda vectorspace ifthefollowingconditions arefulfilled:

1. Addition:If {r1 , r2 } ∈ V and r1 + r2 = r,then r ∈ V

2. Commutativity: r1 + r2 = r2 + r1

3. Associativity: r1 + (r2 + r3 ) = (r1 + r2 ) + r3 and c1 (c2 r) = (c1 c2 ) r

4. Distributivity: c (r1 + r2 ) = c r1 + c r2 and (c1 + c2 ) r = c1 r + c2 r

5. Identityelement: r + 0 = r,1r = r,and r r = r + ( 1) r = 0

Example30 NonorthogonalCoordinateFrame Itispossibletodefineacoordinateframeinwhichthethreescaledlines OA, OB , OC arenonorthogonal.Defining

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