线性代数教学资料-cha课件

2Vectorsin2-spaceand3-space,Overview,Inthischapterwereviewtherelatedconceptsofphysicalvectors,geometricvectors,andalgebraicvectors.Toprovidemaximumgeometricinsight,weconcentrateonvectorsintwo-spaceandthree-space.Later,inChapter3,wewillgeneralizemanyoftheideasdevelopedinthischapterandapplythemtoastudyofvectorsinn-space,thatis,tovectorsinRn.AmajoremphasisinChapter3isoncertainfundamentalideassuchassubspacesofRnandthedimensionofasubspace.AswewillseeinChapter3,conceptssuchassubspaceanddimensionaredirectlyrelatedtothegeometricallyfamiliarnotionsoflinesandplanesinthree-space.,Coresections,VectorsintheplaneVectorsinspaceThedotproductandthecrossproductLinesandplanesinspace,2.1Vectorsintheplane,1.Threetypesofvectors,Physicalvectors:Aphysicalquantityhavingbothmagnitudeanddirectioniscalledavector.Typicalphysicalvectorsareforces,displacements,velocities,accelerations.,(2)Geometricvectors:ThedirectedlinesegmentfrompointAtopointBiscalledageometricvectorandisdenotedby,Foragivengeometricvector,theendpointAiscalledtheinitialpointandBistheterminalpoint.,(3)Equalityofgeometricvectors,Allgeometricvectorshavingthesamedirectionandmagnitudewillberegardedasequal,regardlessofwhetherornottheyhavethesameendpoints.,(4)Positionvectors,(5)Componentsofavector,Theorem2.1.1:Letandbegeometricvectors.Thenifandonlyiftheircomponentsareequal.,(6)AnequalitytestforGeometricVectors,(7)Algebraicvectors:,Theorem2.1.2:Letbeageometricvector,withA=(a1,a2)andB=(b1,b2).Thencanberepresentedbythealgebraicvector,2.Usingalgebraicvectorstocalculatethesumofgeometricvectors,Theorem2.1.2:Letuandvbegeometricvectorswithalgebraicrepresentationsgivenby,Thenthesumu+vhasthefollowingalgebraicrepresentation:,3.Scalarmultiplication,Theorem2.1.3:Letubeageometricvectorswithalgebraicrepresentationsgivenby,Thenthescalarmultiplecuhasthefollowingalgebraicrepresentation:,4.Subtractinggeometricvectors,5.ParallelvectorsVectorsuandvareparallelifthereisanonzeroscalarcsuchthatv=cu.Ifc0,wesayuandvhavethesamedirectionbutifc0,wesayuandvhavetheoppositedirection.,6.Lengthsofvectorsandunitvectors,7.Thebasicvectorsiandj,2.1ExerciseP12626,2.2Vectorsinspace,1.Coordinateaxesinthreespace,2.Theright-handrule,3.Rectangularcoordinatesforpointsinthreespaceaxis;coordinateplanes;octants,4.Thedistanceformula,Theorem2.2.1:LetP=(x1,y1,z1)andQ=(x2,y2,z2)betwopointsinthreespace.ThedistancebetweenPandQ,denotedbyd(P,Q),isgivenby,5.Themidpointformula,Theorem2.2.2:LetP=(x1,y1,z1)andQ=(x2,y2,z2)betwopointsinthreespace.LetMdenotethemidpointofthelinesegmentjoiningPandQ.Then,Misgivenby,6.Geometricvectorsandtheircomponents,7.Additionandscalarmultiplicationforvectors,8.Parallelvectors,lengthsofvectors,andunitvectors,9.Thebasicunitvectorsinthreespace,2.3Thedotproductandthecrossproduct,1.Thedotproductoftwovectors,Definition2.3.2:Letuandvaretwo-dimensionalvectors,thenthedotproductofuandv,denoteduv,isdefinedbyuv=u1v1+u2v2.Letuandvarethree-dimensionalvectors,thenthedotproductofuandv,denoteduv,isdefinedbyuv=u1v1+u2v2+u3v3.,Definition2.3.1:Letuandvarevectors,thenthedotproductofuandv,denoteduv,isdefinedbyuv=|u|v|cos.whereistheangleofvectorsuandv.,2.Theanglebetweentwovectors,uv=|u|v|cos.,3.Algebraicpropertiesofthedotproduct,4.OrthogonalVectors(正交向量）When=/2wesaythatuandvareperpendicularororthogonal.,Theorem2.3.1:Letuandvarevectors,thenuandvareorthogonalifandonlyifuv=0.,Intheplane,thebasicunitvectorsiandjareorthogonal.Inthreespace,thebasicunitvectorsi,jandkaremutuallyorthogonal.,5.Projections,6.Thecrossproduct,Definition2.3.3:Letuandvarevectors,thenthecrossproductofuandv,denoteduv,isavectorthatitisorthogonaltouandv,andu,v,uvisright-handsystem,andthenormofthevectoris|uv|=|u|v|sin.whereistheangleofvectorsuandv.,Unitvector,direction,7.Remembertheformofthecrossproduct(twomethods),determinant,8.Algebraicpropertiesofthecrossproduct,9.Geometricpropertiesofthecrossproduct,10.Tripleproducts（三重积）,11.Testsforcollinearityandcoplanarity,2.3ExerciseP14848,Theorem:Letu,vandwbenonzerothreedimensionalvectors.uandvarecollinearifandonlyifuv=0.u,vandwarecoplanarifandonlyifu(vw).,2.4Linesandplanesinspace,1.Theequationofalineinxy-plane,y,2.Theequationofalineinthreespace,(1)Pointanddirectionalvectorformequationofaline,(2)Parametricequationsofaline,Example1:LetLbethethroughP0=(2,1,6),havingdirectionvectorugivenbyu=4,-1,3T.FindparametricequationsforthelineL.DoesthelineLintersectthexy-plane?Ifso,whatarethecoordinatesofthepointofintersection?,Example2:FindparametricequationsforthelineLpassingthroughP0=(2,5,7)andthepointP1=(4,9,8).,2.Theequationofaplaneinthreespace,Pointandnormalvectorformequat