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MATH 1229A Whole Lecture Notes.pdf

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Department
Mathematics
Course
Mathematics 1229A/B
Professor
Prof
Semester
Fall

Description
Math1229ABUnit1VectorstextreferenceSection11cVOlds2010Unit111VectorsYouarefamiliarwiththesetofrealnumbersRealnumbersmeansallthenumbersyouveeverheardoforcanimagineunlessyouvelearntaboutoratleastheardofimaginarynumberstheyrenotrealnumbersRealnumbersincludesalltheintegersincludingbothpositiveandnegativeandofcourse0fractionscalledrationalnumbersdecimalsthatcantbeexpressedasfractionstheirrationalnumbersAllthenumbersalongtherealnumberlinefromtoWecallthesetofrealnumbersYourealsofamiliarwiththexyplaneYouvedrawngraphsinthisplaneTheresthexaxishorizontalandtheyaxisverticalwhichcrossattheoriginEachaxisisbasicallyarealnumberlineAnypointinthisplanecanbeexpressedasanorderedpairxygivingitsxcoordinateanditsycoordinateEachcoordinatecanbeanyrealnumberWecallthexyplane2spaceandthe2setcontainingallofthepointsinthisplaneiscalledWepronouncethatR2ieArtoo2SocanbethoughtofasthesetofallorderedpairsofrealnumbersYouveprobablyalsoseensomethingevenmorecomplicatedwith3axesthexaxistheyaxisandthezaxisEachaxisisperpendiculartobothoftheotherswhichmakesithardtodrawonapieceofpaperorablackboardTheyreoftendrawnwiththeyaxishorizontalthezaxisverticalandthexaxisoatafunnyangletorepresentthatitscomingstraightoutofthepageatyouThe3axesrepresentthe3dimensionsofspaceierealityLiketheroomyouresittinginTheresnotonlyupanddownandleftandrightbutalsonearandfarorhereandthereorwellyouknowthatthirddimensionwhichwemightcalldepthInMathwecalltheregiondenedbythese3axes3spaceAndpointsin3spacearerepresentedbyorderedtriplesxyzgivingthexyandzcoordinatesofthepointAsbeforeeachofthesecoordinatescanbeanyrealnumberThe3setcontainingallofthepointsin3spaceiscalledpronouncedR3andwecanthinkofthisasthesetofallorderedtriplesofrealnumbersNowthatyouvegotthatstraightletsconfusethingsSometimeswhenwewriteanorderedpairoranorderedtripleitdoesntrepresentapointInsteaditrepresentsadirectedlinesegmentcalledavectorAndwhenthepairortriplerepresentsavectorthenumbersinitorsymbolsrepresentingnumbersarentcalledcoordinatestheyrecalledcomponentsThatseemslikeitwillbeconfusingusingthesamenotationtodenotetwodierentthingsItsnottoobadthoughbecauseyoucantellbythecontextwhetherapairortripletisapointoravectorAndbecausewewritethenamesdierentlydependingwhetheritsapointoravectorPointsarenamedwithnormalcapitallettersusuallyPorsomethingnearbyinthealphabetandsometimeswithsubscriptsSowemighthavethepointsPppandQqqorthepoints1212PxyzandPxyzWithvectorswedontusecapitallettersandwedosomething11112222toshowthatitsavectorInourtextbooktheyuseboldfacetypeforthenameofavectorForinstancethevectorvvvorthevectoruuuuSureIhearyouthinkingthats12123easyenoughforyoubutwhataboutmewhenImwritingwithapenorpencilWellthatswhyweregoingtouseadierentconventioninthesenotesOnethatsmuchmoreobviousWhenaletteristhenameofavectorwellputanarrowoveritLikethisvvv12SowhyisitthatwerepresentavectorwhichwesaidisadirectedlinesegmentusingsomethingthatlookslikeapointWellitsjustaconventionItsshorthandWhenwesayv12whatwemeanisthatthevectorvisthedirectedlinesegmentthatstartsattheoriginandendsatthepoint12Soheresapictureofthevectorv12Itstartsatthepoint00andthenendsataplacethats1unittotherightand2unitsupfromtherey632v10123x2Unit12DenitionThevectorvvvinisthedirectedlinesegmentthatgoes12fromtheoriginiethepoint00tothepointVvvSimilarlythevector123vvvvinisthedirectedlinesegmentthatgoesfromthepoint000to123thepointVvvvThepointViscalledtheendpointofthevectorv1232Ifwewanttosaythatvisavectorinieavectorwhichhastwocomponentsthenwesay2vYouveprobablyseenthatsymbolbeforeforinstanceforsayingthataparticularobject33isanelementofaparticularsetSimilarlywecansayvtostatethatvisavectorin2NoticethatearlierwesaidthatwasthesetofallpointsinthexyplaneButnowweresaying2thatavectorisinthatsetHmmIfisasetdoesitcontainpointsorvectorsWellactuallywecanthinkofiteitherwayWecandescribe2spaceasthesetofallpointsinthexyplaneorasthesetofallvectorsinthexyplaneOrwecandeneitsimplyasthesetofallorderedpairsxyThatsprobablythebestwaytothinkofitBecauseanorderedpaircanaswevealreadydiscussedrepresenteitherapointoravectordependingonthecontextLikewisewellthinkof3assimplythesetofallorderedtriplesxyz2ThereisavectorincorrespondingtoeachpointinthexyplaneLikewisethereisavector3incorrespondingtoeachpointin3spaceButwaitaminuteWhataboutthepoint00or000ThatswherethedirectedlinesegmentstartsSocantherebeadirectedlinesegmentthatgoesfromthatpointtoitselfWellyesAlthoughyouwontbeabletoseeitandyouwontcareorbeabletodeterminewhatdirectionitgoesinThatisinspiteofthefactthatthelinesegmentfromthepoint00tothepoint00seemsnonsensicalbecausetheresnolinesegmenttherewedoconsidertheretobeavectorv00ItactuallycomesinveryhandyWecallitthezerovectorandgiveitthename0DenitionAzerovectorisavectorwhosecomponentsareall0Thezerovector23inisthevector000Similarly0000isthezerovectorinWheneverwedeneanewmathematicalconstructweneedtodenewhatequalitymeansforthatconstructEvenifitseemsprettyobviousSoweneedtodenewhatitmeanstosaythattwovectorsareequalWevealreadyusedthatconceptinattachingnamestovectorsForinstance2whenwesayvvvweresayingthatthevectorwhosenameisvisequaltothevectorin12whoserstcomponentisvandwhosesecondcomponentisvLikewisewhenwesay0000123weresayingthatthevectorwhosenameis0isequaltothevectorinwhosecomponentsareall0ButofcoursewhatwereallymeantthereisthisisthenameImgoingtocallthatvectorbyratherthanhereare2dierentvectorsandtheyreequalButoftenwedoneedtoequate2vectorsinthatsensetooOrtosaythatthevectoryougetwhenyoudocertainvectorarithmeticoperationswhichwelllearnaboutshortlyisequaltoaspeciedvectorSowhatdowemeanwhenwesayforinstancethatuvDenitionTwovectorsareequaliftheyarevectorsinthesamespaceandtheir2correspondingcomponentsareequalThatistwovectorsinareequaliftheyhavethesamerstcomponentandalsohavethesamesecondcomponentSimilarlytwo3vectorsinareequaliftheyhavethesamerstcomponentandhavethesamesecondcomponentandhavethesamethirdcomponentInmathematicalnotationwehaveIfuuuandvvvthenuvifandonlyifuvanduv12121122andsimilarlyIfuuuuandvvvvthenuvifandonlyifuvanduvanduv1231231122332andNoticethatinorderfor2vectorstobeequaltheymustbevectorsinthesamespaceIfu3vthentheycanneverbeequalvectorsnomatterwhattheircomponentsare
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