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Unit 3: Lines and Planes

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Department
Mathematics
Course
Mathematics 1229A/B
Professor
Allen O' Hara
Semester
Fall

Description
Math1229ABUnit3LinesandPlanestextreferenceSection13cVOlds201030Unit33LinesandPlanes2LinesinYouarealreadyfamiliarwithequationsoflinesInpreviouscoursesyouwillhavewrittenequationsoflinesinslopepointforminslopeinterceptformandprobablyalsoinstandardform2foralineinRecallthatyymxxistheslopepointformequationofthelinethroughpointxywithslopem1111ymxbistheslopeinterceptformequationofthelinewithslopemandyinterceptbaxbycisthestandardformequationwhicheitheroftheotherscanberearrangedtoInthiscoursewedontusetheslopepointorslopeinterceptformsofequationsoflinesInsteadweusevariousothervectorbasedformsofequationsButwedostillusethestandardform23Youalreadyknowthatgivenany2distinctpointswhetherinorinthereisexactlyonelinewhichpassesthroughbothpointsSupposewehave2pointsPandQLetbethelinethatpassesthroughthesetwopointsBothpointPandpointQlieonlineandsodoallthepointsbetweenthemInfactthedirectedlinesegmentPQliesonlineWhenthisdirectedlinesegmentistranslatedtotheorigintheresultingvectormostlikelydoesntlieonlineunlesstheoriginhappenstolieonlinebutifnotitdoeslieonalinewhichisparalleltolineItliesonthelineparalleltowhichpassesthroughtheoriginSothisvectordoesgiveussomeinformationabout2thelineSimilartotheinformationgivenbyknowingtheslopeofalineinalthoughitsnotquitethesameinformationIfavectorvliesonaparticularlineoronalineparalleltothatlinewesaythatvisparalleltooriscollinearwiththatlineAndwecallvadirectionvectorforthelineNotthatthelineactuallyhasadirectionassociatedwithitItdoesntItextendsinbothdirectionsbuthasnoparticularforwardsalongthelineorbackwardsalongthelineassociatedwithitSodontreadtoomuchmeaningintothetermdirectionvectorIfvisadirectionvectorforlinethensoisvAndsoiseveryotherscalarmultipleofvexceptfor0vBecauseofcourse0v0whichhasnodirectioninformationButeveryotherscalarmultipleofvstartsattheoriginandgoeseitherthesameortheoppositedirectionasvandthereforealsoliesonthelineparalleltowhichpassesthroughtheoriginSoanysuchvectorwouldbeconsideredadirectionvectorforlineDenitionAnynonzerovectorwhichisparalleltoalineiscalledadirectionvectorforlineForinstanceconsiderthelinexy2Thepoints11200213312442etcalllieonthislineSodo1232and3212andinnitelymanyotherpointsPickany2ofthesepointsandndthevectorwhichisthetranslationtotheoriginofthedirectedlinesegmentbetweenthemandyouhaveadirectionvectorforthelineAndweknowthatforanypointsPandQlettingpOPdenotethevectorfromtheorigintopointPandqOQdenotethevectorfromtheorigintopointQthevectorvqpisthetranslationofdirectedlinesegmentPQtotheoriginSoforinstanceforpointsP11andQ02wehavep11andq02andweseethatvqp021111isadirectionvectorforthelinexy2AndotherchoicesofPandQgiveotherdirectionvectorswhicharescalarmultiplesofthisoneGoaheadpicksomeotherpointsandseewhatvectorsyouget
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