《高数双语》课件section 8.2

Section8.2ProductsofVectors1Euclid2TheDotProduct(点积,数量积,内积)ofTwoVectorsThedotproductcanbeusedtoexpresstheworkdonebyagivenforce.Letaandbbetwovectors,andsupposeθistheangleDefinitionThentherealnumberbetweenaandb,denoteby

iscalledthedotproduct(scalarproduct,innerproduct)

ofaandb,denotedbythatis,or3TheBasicPropertiesofDotProductand(2)Nonnegativity:(3)Commutativelaw:(5)Associativelawwiththescalarmultiple:(4)Distributivelaw:Thedotproductshavethefollowingbasicproperties:(1)4TheComponentRepresentationofTheDotProductSincethebasicunitvectorsi,j,kareperpendiculartooneanother,usingthedefinitionofthedotproductwehaveIfandandthenExamples(1)(2)5SomeApplicationsofTheInnerProductinGeometry(1)Thenorm(模，范数)ofavectorBythedefinitionofthenormofavector,wehave(2)Theincludedangle(夹角)betweentwononzerovectorsLetaandbbetwononzerovectors.Wehave6SomeApplicationsofTheInnerProductinGeometry(3)TheProjection(投影)thenwehaveSinceandandtheprojectionvectorofaontobisthereforetheprojectionvectorofbontoais7SomeApplicationsofTheInnerProductinGeometryExampleandSupposethat(1)Find(2)Findtheanglesbetweenaandb(3)Findtheprojectionofaontob.SolutionFinish.SomeApplicationsofTheInnerProductinGeometry8ExampleandForthepointsprovethatthelinethroughAandBisperpendiculartothelinethroughCandD.SolutionSo,thelinethroughAandBisperpendiculartothelinethroughCandD.Finish.9SomeApplicationsofTheInnerProductinGeometryExampleProvetheCauchyinequality:LetProofandLetthevectorSincewehaveBythecomponentrepresentationofdotproduct,wewillobtaintheCauchyinequality.Finish.10TheVectorProductofTwoVectorsinSpaceWestartwithtwononzerovectorsuandvinspace.Ifuandvarenotparallel,theydetermineaplane.Weselectaunitvectornperpendiculartotheplanebytheright-handrule.Thismeansthatwechoosentobetheunit(normal)vectorthatpointsthewayyourrightthumbpointswhenyourfingerscurlthroughtheangleθfromutov.canbedefinedThenthevectorproductasfollowing.11Definition

Vector(Cross,outer)Product(向量积,叉积,外积)TductofuandvbecauseofthecrossinthenotationisorthogonaltobothuandvbecauseitisscalarThevectorSincethesinesof0andπarebothzero,itmakessensetodefinethecrossproductoftwoparallelnonzerovectortobe0.tobezero.Ifoneorbothofuandvarezero,wealsodefineTheVectorProductofTwoVectorsinSpace12TheVectorProductofTwoVectorsinSpaceProofTwovectorsaandbareparallel(orcollinear)ifandonlyifWeassumethataandbarebothnonzerovectors.Ifa=0(orb=0),thenthisconclusionobviouslyholds.orTheoremthatis,Twovectors

aandbareparallel(orcollinear)ifandonlyifWehavethat,13PropertiesoftheVectorProductIfu,vandwareanyvectorsandr,sarescalars,then(1)Anti-commutativelaw(3)Distributivelaw(2)Associativelawwithrespecttothescalarmultiple14TheComponentRepresentationoftheVectorProductThenthedistributivelawsandtherulesformultiplyingi,j,andSupposethatktellusthat15TheComponentRepresentationoftheVectorProductThetermsinthelastlinearethesameasthetermsintheexpansionofthesymbolicdeterminant16TheComponentRepresentationoftheVectorProductSolutionIfandandfindExampleFinish.17TheComponentRepresentationoftheVectorProductSolutionLetθbetheanglebetweenthevectorsandfindExample

Finish.18TheGeometricMeaningoftheNormoftheVectorProductisBecausenisaunitvector,themagnitudeofThisistheareaoftheparallelogramdeterminedbyuandv,|u|beingthebaseoftheparallelogramandtheheight.19TheComponentRepresentationoftheVectorProductSolutionFindtheareaofthetrianglewhoseverticesareSincebparallelogramwithadjacentsidesABandAC,thatisTheareaSofthetriangleABCishalftheareaoftheExample

Finish.Wehave20VectorProductExample

FindthelinespeedvectorvatanypointPonthebody.SolutionWetakeavectorwintheaxisl,suchanditspositivedirectionisdeterminedthatLetObeapointontheaxisbytheright-handrule.l,thenthelinespeedLetThelinespeedvectorvthenisperpendiculartobothwandr,andw,r,vsatisfiesThereforetheright-handrule.Arigidbodyrotatesaroundafixedaxislwithangularvelocityw.Finish.21TorqueWhenweturnaboltbyapplyingaforceFtoawrench,thetorqueweproduceactsalongtheaxisofthebolttodriveThemagnitudeofthetorquetheboltforward.dependsonhowfaroutonthewrenchtheforceisappliedandonhowmuchoftheforceisperpendiculartothewrenchattheThenumberweusetopointofapplication.measurethemagnitudeistheproductofthelengthoftheleverarmrandthescalarcomponentofFperpendiculartor.22TorqueIntherightfigure,wehaveItiswellknownthatiftheforceFisparalleltothewrench,thetorqueproducediszero.MagnitudeoftorquevectororIfweletnbeaunitvectoralongtheaxisoftheboltinthedirectionofthetorque,thenacompletedescriptionoftheorTorquevectortorquevectoris23TripleScalarorBoxProductDefinition

TripleScalarorBoxProduct

Supposeu,vandwarethreevectors,thentheproductiscalledthetriplescalarproductofu,vandw.Sometimesthetriplescalar