高数英文版

Lecture5PlanesandLinesinSpace2EquationsforPlanesinSpaceAplaneinspaceisdeterminedbyknowingapointontheplaneandits“tilt”ororientation.(,,)PxyzThenMisthesetsofallpointsSupposethatplaneMpassesnijk.ABCnonzerovectorandisnormal(perpendicular)tothethroughapoint0000(,,)Pxyz0PPforwhichisorthogonalton.Thus,thedotproduct0n0.PPThisequationisequivalentto000(ijk)()i()j()k0ABCxxyyzzor000()()()0.AxxByyCzzPointnormalformscalarequationoftheplane3EquationsforPlanesinSpaceEquationforaplanehas0000(,,)PxyzTheplanethroughnijkABCnormaltoComponentequationsimplified:0n0PPVectorequation:000()()()0AxxByyCzzComponentequation:,AxByCzDwhere000DAxByCzTwoplanesareparallelifandonlyiftheirnormalsareparallel,or12nnkforsomescalark.4FindinganEquationforaplaneperpendicularto0(3,0,7)PFindanequationfortheplanethroughn5i2jk.SolutionThecomponentequationis5152705222.xyzxyzSimplifying,weobtain5((3))2(0)(1)(7)0.xyz5FindinganEquationforaPlanethroughThreePoints(0,3,0)C(0,0,1)A(2,0,0)BFindanequationfortheplanethroughand(0,0,1),A(2,0,0)B(0,3,0).CnSolutionIWehavetofindavectornormaltotheplaneanduseitwithoneofthepoints(itdoesnotmatterwhich)towriteanequationfortheplane.Thecrossproductijk201031ABACxyzO3i2j6kn.6FindinganEquationforaPlaneThroughThreePointsSolution(continued)(0,3,0)C(0,0,1)A(2,0,0)BxyznItiseasytoseethatnisnormaltotheplane.Wesubstitutethecomponentsofthisvectorinto(0,0,1)Aandthecoordinateofthepoint-normalformoftheequationandobtain3(0)2(0)6(1)0xyz3266.xyz000()()()0.AxxByyCzz7SolutionIIisanypointintheplane,Supposethat(,,)Pxyz(0,3,1)AC(,,1)APxyz(2,0,1)ABSincethesethreevectorarecoplanarifandonlyifthepointPliesintheplane,,,0,APABAC(0,3,0).C(0,0,1),A(2,0,0)BxyzO(,,)PxyzthensowehaveFindinganEquationforaPlaneThroughThreePoints8Solution(continued)(0,3,0).C(0,0,1),A(2,0,0)Bthatis12010.031xyzExpandedthedeterminantontheleftsideofaboveequation,wehave3266.xyzxyzO(,,)PxyzFindinganEquationforaPlaneThroughThreePoints9InterceptFormoftheEquationforaPlaneIngeneral,iftheinterceptsoftheplanewiththex-axis,y-axisandz-axis(0,,0)Bb(0,0,)Cc(,0,0)Aafortheplane1.xyzabcand,OAaOBbbe,OCcThen,justasthelastrespectively.example,wecanobtaintheequationxyzO(,,)PxyzInterceptformoftheequation10GeneralEquationsforPlanesGeneralEquationforaplaneTheequationcanberewrittenintheform,AxByCzDwhere000DAxByCzTherefore,theequationofanyplaneisalinearequationinthreeConversely,anylinearequationinthreevariablesrepresentsvariables.ifA,B,Carenotall0.n(,,),ABCaplanewithnormalvectortheequationcanbewrittenas0,CInfact,if(0)(0)0.DAxByCzC11SomePlaneswithSpecialLocations(1)Ifagivenplanepassesthroughtheorigin(0,0,0),Othen0xyzsatisfythegeneralequationfortheplane,sothat0.DTherefore,theequationoftheplanethroughtheoriginis0.AxByCzOxyz12OxyzSomePlaneswithSpecialLocations(2)Ifagivenplaneisparalleltothez-axis,andnk0.CTherefore,theequationofthisplaneisn(,,)ABCk(0,0,1),isorthogonalto0.AxByD0,0,ByCzDAxCzDSimilarly,theequationsofplaneswhichareparalleltothex-axisory-axisarerespectively.nthenthenormalvector13OxyzSomePlaneswithSpecialLocations(3)Ifagivenplaneisorthogonaltothez-axis,0.ABTherefore,theequationofthisplaneis0CzD0,0,AxDByDSimilarly,theequationsofplaneswhichareorthogonaltothex-axisory-axisarerespectively.andso0.DzzCorn0zn//kthen14SomeProblemsaboutPlanes(1)AngleBetweenPlanes11n2nTheanglebetweentwointersectingplanesisdefinedtobethe(acute)angledeterminedbythenormalvectorsasshowninthefigure.Let22222:0.AxByCzD11111:0,AxByCzD2and1111n(,,)ABCTherenormalvectorscanbechosenasThen2222n(,,),ABCrespectively.1212121222222212111222||nn||||cos.||n||||n||AABBCCABCABC15Iftwoplanesareparallelororthogonal,theirnormalvectoralsoparallelororthogonal.21212120.AABBCC1212nn2111222.ABCABC12//12n//n12n1n12n1nSomeProblemsaboutPlanes(2)PositionRelationshipsBetweenTwoPlanes16Wechoosearbitrarilyagivenpointwhichdoesnotlieintheplane.beagivenplane,:0AxByCzDLet1P0Panddrawapointintheplane1111(,,)Pxyzavector10.PPpointtotheplaneisequaltotheabsolutevalueoftheprojectionofthevectorThenthedistancefroma10PPontothenormalvectornof.ndThus,bytheformulaofprojection,weget10|n|.dPPbe0000(,,)PxyzandSomeProblemsaboutPlanes(3)DistanceFromaPointtoaPlane17TheDistancefromaPointtoaPlane000222||.AxByCzDdABCSinceThereforeThustheformulaofthedistanceis10|n|dPPand2221n(,,).ABCABC10010101(,,)PPxxyyzzliesintheplane,1PNoticethat1110.AxByCzD010101222|()()()|.AxxByyCzzdABC1P0Pndsothat18SomeExamplesaboutPlanesExample1.Discussthepositionrelationshipsbetweenthefollowingplanes:12(1):210,:310xyzyzSolution(1)Since22222|102113|cos(1)2(1)13thenthesetwoplanesintersectandtheanglebetweenthemis1.601arccos.601912(2):210,:42210xyzxyzSolution(2)Since1a{2,1,1},2a{4,2,2}and211,422thenthesetwoplanesareparallel.Again,since1(1,1,0)Mbut2(1,1,0),Mthesetwoplanesarenotsame.Example1.Discussthepositionrelat