大一各种试题高等数学下袁建华lecture

b SectionVectorsin andSpacebSectionVectorsin andSpaceConceptsConceptsofLinearOperationonVectorsRectangularCoordinateSystemVectorsin neandSpaceNorm,DirectionAnglesandProjectionofDefinitionDefinitionVectorsandtiesthatpossessbothPTerminalDefinitionThemagnitudeofavectoriscalledthelengthnorm模)ofthe-DefinitionTwovectorsareequal(相等的)orthesameiftheyhavethesamemagnitudeanddirection.DefinitionThenegativevector(负矢量)isavectorwhichhasthesamemagnitudewiththeoriginalb,andhasanoppositedirectionwithb,denotedby–b.DefinitionIftwovectorsaandbhavesameDefinitionIftwovectorsaandbhavesameoroppositedirection,wesaythattheyareparallel(平行)orcollinear(共线),denotedbya//b.DefinitionIfthedirectionsoftwovectorsaandbareorthogonal,wesaythattheyareorthogonal(正交)orperpendicular(垂直),denotedbyab.DefinitionSupposethata1,a2,…,ak(k>2)arekvectorswithacommoninitialpoint.Iftheylineinthesamene,thenwesaythatthesevectorsareconar(共面).DefinitionAvectorwhoselengthis1iscalledaunitvector(单位向量),aunitvectorwhosedirectionisthesameasthatofaiswrittenasa.DefinitionAvectorwhoselengthis0iscalledthezero零向量 2.Operations2.OperationsonDefinitionTrianglelawofadditionofvectors(向量加法的三Supposeaandbaretwovectors.Ifwedrawavector,whichisequaltob,fromtheinitialpointofa,thenthesuma+bofaandbisthevectorextendingfromtheterminalpointofatotheinitialpointofb.2.OperationsonParallelogramlawofadditionofvectors(向量加法的平行四OperationsOperationsonTheadditionofvectorssatisfiesthefollowing2.OperationsonDefinitionThedifference(减法)oftwovectorsaandbisgivenbyaba(b). -DefinitionScalarMultiplication(数乘向量Theproductofascalarmandavectoraexpressedbymaisavector.Itslengthis|ma|.ItsdirectionisthesameasthatofaifmispositiveandisoppositetothatofaifmisOperationsonThescalarproductsatisfythefollowing OperationsonThescalarproductsatisfythefollowingOperationson Absolute |a||||aAdditionandscalarproductofvectorsarecalledbyajointlinearoperation(线性运算onThree nes(坐标面):xOy,yOzandEightoctants卦限position(坐标分解式)4.Vectors4.VectorsneandyComponentRepresentationofLinearOperationsonVectorsBymeansoflinearoperationsonvectorswea(x1)i(y1)j(z14.VectorsExampleneand4.Vectors4.Vectorsneand4.VectorsneandExample2GivenpointsP(2,5,6)andQ(6,9,-2),pleasefindpointM(x,y,z)onthelinesegmentPQsuchthatPMQ(6,9,-4.Vectors4.Vectorsneand5.Length,DirectionAnglesandDirectioncosinesofa DirectioncosinesofaDirectioncosinesofaDirectioncosinesofaObviously,thedirectionofavectoraisdeterminedcompleybythedirectioncosinesofthevector.ence,todeterminethedirectionofavectora,weneedonlydeterminethedirectioncosinesofDirectionDirectioncosinesofaBythelastformula,itiseasytoseethefollowingThequadraticsumofthedirectioncosinesofanynonzerovectorisequalto1,thatiscos2cos2cos2Thethreecomponentsoftheunitvectoraarejustitsthreedirectioncosines,thataoax,y,z|a||a||a||a|FindtheLengthandDirectionCosinesofaVector We Weknowthatthedirectionofavectorcanbedeterminedbyitsdirectioncosines.Butsometimesweneedonlytoconsidertheazimuth(方位ofthevectorandarenotinterestedinitssense(指向)(vectoraand-ahavethesameazimuthbutoppositesense).DirectionIfweneedonlytoconsidertheazimuthandarenotinterestedintheInthiscase,weneedonlyknowthreenumbersproportionaltothedirectioncosinesanditisnotnecessarytofindthedirectioncosinesofthevector.DirectionA2DirectionA2B2CA2B2CA2B2CDirectionA2B2CA2B2CA2B2CApplicationApplicationoftheDirectionofaa, ,AnglebetweentwoTheProjectionofTheProjectionof TheProjectionofTheProjectionofTheProjectionof(b)a|b|SectionSectionDotProductTripleScalarTheTheDotProduct(点积,数量积,内积ofTwoVectors Letaandbbetwovectors,andsupposeθistheanglebetweenaandb,denoteby (a,b).Thentherealnumber|a||b|iscalledthedotproduct(scalarproduct,innerproduct)ofaandb,denotedby a•b,thatis, a•b|a|(b)a|b|(a)bThedotproductcanbeusedtoexpresstheworkdonebyagivenTheBasicPropertiesofDotThedotproductshavethefollowingbasicCommutativeAssociativelawwiththescalarab(ab(axiayjazk)(bxibyjbzk)axbxaybyazbzTheComponentRepresentationofDotSomeApplicationsofTheInnerinSomeSomeApplicationsofTheInnerProductinGeometryprojba•b(b•a)jaa•b(a•b)bSomeApplicationsofTheInnerProductinGeometrySomeSomeApplicationsofTheInnerProductinGeometrySomeApplicationsofTheInnerProductinGeometrySomeApplicationsofTheInnerProduct|a•bSomeApplicationsofTheInnerProduct|a•b||a||b|TheVectorProductofTwoVectorsTheTheVectorProductofTwoVectorsinDefinitionVector(CrossouterProduct(向量积,叉积,外积TheVectorProductofTwoVectorsinTwovectorsaandbareparallel(orcollinear)ifandonlyProofIfa=0(orb=0),thenthisconclusionobviouslyWeassumethataandbarebothnonzero Twovectorsandbareparallel(orcollinear)ifandonlyifthatis,WehavePropertiesPropertiesoftheVectorIfu,vandwareanyvectorsandr,sarescalars,mutativeAssociativelawwithrespecttothescalarTheComponentRepresentationoftheVectorProductSupposeThenthedistributivelawsandtherulesformultiplyingi,j,klusTheComponentRepresentationofVectorThetermsinthelastlinearetheTheComponentRepresentationofVectorThetermsinthelastlinearethesameasthetermsintheexpansionofthesymbolicdeterminant u1u2u3v1v2uv(u2v3u3v2)i(u1v3u3v1)j(u1v2u2v1TheComponentRepresentationofVectorTheTheComponentRepresentationoftheVectorProductBecausenisaunitvector,the