AdvancedDynamics
RezaN.Jazar
SchoolofAerospace,Mechanical,andManufacturingEngineering
RMITUniversity Melbourne,Australia
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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
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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.
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
