Introduction to vectors


 Delilah Lawson
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1 Introduction to vectors mctyintrovector Avectorisquntityththsothmgnitude(orsize)nddirection. Bothofthese propertiesmustegiveninordertospecifyvectorcompletely.inthisunitwedescriehowto writedownvectors,howtoddndsutrctthem,ndhowtousethemingeometry. In order to mster the techniques explined here it is vitl tht you undertke plenty of prctice exercises so tht they ecome second nture. Afterredingthistext,nd/orviewingthevideotutorilonthistopic,youshouldeleto: distinguish etween vector nd sclr; understnd how to dd nd sutrct vectors; knowwhenonevectorismultipleofnother; use vectors to solve simple prolems in geometry. Contents 1. Introduction 2 2. Representing vector quntities 2 3. Position vectors 3 4. Some nottion for vectors 3 5. Adding two vectors 4 6. Sutrcting two vectors 5 7. Addingvectortoitself 5 8. Vectors of unit length 6 9. Using vectors in geometry c mthcentre 2009
2 1. Introduction Vector quntities re extremely useful in physics. The importnt chrcteristic of vector quntityisthtithsothmgnitude(orsize)nddirection. Bothofthesepropertiesmuste given in order to specify vector completely. Anexmpleofvectorquntityisdisplcement.Thistellushowfrwywerefromfixed point,nditlsotellsusourdirectionreltivetothtpoint. P O Another exmple of vector quntity is velocity. This is speed, in prticulr direction. An exmpleofvelocitymighte60mphduenorth. Aquntitywithmgnitudelone,utnodirection,isnotvector. Itisclledsclrinsted. Oneexmpleofsclrisdistnce.Thistellsushowfrwerefromfixedpoint,utdoesnot giveusnyinformtionoutthedirection.anotherexmpleofsclrquntityisthemssof n oject. Key Point A vector hs oth mgnitude nd direction, nd oth these properties must e given in order tospecifyit.aquntitywithmgnitudeutnodirectionisclledsclr. 2. Representing vector quntities Wecnrepresentvectorylinesegment.Thisdigrmshowstwovectors. B A Wehveusedsmllrrowtoindictethtthefirstvectorispointingfrom Ato B.Avector pointing from B to A would e going in the opposite direction. Sometimeswerepresentvectorwithsmlllettersuchs,inoldtypefce. Thisis common in textooks, ut it is inconvenient in hndwriting. In writing, we normlly put r underneth,orsometimesontopof,theletter: or.inspeech,wecllthisthevector r. 2 c mthcentre 2009
3 3. Position vectors Sometimesvectorsrereferredtofixedpoint,norigin. Suchvectorisclledposition vector. Sowemightrefertothepositionvectorofpoint P withrespecttonorigin O. In writing, might put OP for this vector. Alterntively, we could write it s r. These two expressions refer to the sme vector. P O r 4. Some nottion for vectors Whtdoesitmenif,fortwovectors, =?Thismensfirstthtthelengthof equlsthe lengthof,sothtthetwovectorshvethesmemgnitude.butitlsomenstht nd reinthesmedirection.howcnwewritethisdownmoresuccinctly? Iftwovectorsre inthesmedirection,thentheyreprllel.wewritethisdowns //. Forlength,ifwehvevector AB,wecnwriteitslengths ABwithoutther.Alterntively, wecnwriteits AB.Thetwoverticllinesgiveusthemodulus,orsizeof,thevector.Ifwe hvevectorwrittens,wecnwriteitslengthseither withtwoverticllines,ors inordinrytype(orwithoutther).thisiswhyitisveryimportnttokeeptotheconvention ththseendoptedinordertodistinguishetweenvectornditslength. Key Point Thelengthofvector ABiswrittens ABor AB, ndthelengthofvector iswrittens (inordinrytype,orwithoutther)ors. Iftwovectors nd reprllel,wewrite // 3 c mthcentre 2009
4 5. Adding two vectors Oneofthethingswecndowithvectorsistoddthemtogether. Weshllstrtydding twovectorstogether. Oncewehvedonetht,wecnddnynumerofvectorstogethery ddingthefirsttwo,thenddingtheresulttothethird,ndsoon. Inordertoddtwovectors,wethinkofthemsdisplcements. Wecrryoutthefirstdisplcement, nd then the second. So the second displcement must strt where the first one finishes. + Thesumofthevectors, + (ortheresultnt,sitissometimesclled)iswhtwegetwhen wejoinupthetringle.thisisclledthetringlelwforddingvectors. There is nother wy of dding two vectors. Insted of mking the second vector strt where thefirstonefinishes,wemkethemothstrttthesmeplce,ndcompleteprllelogrm. This is clled the prllelogrm lw for dding vectors. It gives the sme result s the tringle lw,ecuseoneofthepropertiesofprllelogrmisthtoppositesidesreequlndinthe smedirection,sotht isrepetedtthetopoftheprllelogrm. + Key Point Wecnddtwovectors nd ymking strtwhere finishes, ndcompletingthe tringle.alterntively,wecnmke nd strttthesmeplce,ndtkethedigonlof the prllelogrm. 4 c mthcentre 2009
5 6. Sutrcting two vectors Whtis?Wethinkofthiss + ( ),ndthenweskwht mightmen.thiswill evectorequlinmgnitudeto,utinthereversedirection. Nowwecnsutrcttwovectors.Sutrcting from willethesmesdding to. Key Point mens + ( ) 7. Adding vector to itself Wht hppens when you dd vector to itself, perhps severl times? We write, for exmple, + + = 3. Inthesmewy,wewouldwrite n = } + {{... + }. ncopies 5 c mthcentre 2009
6 Key Point Avector nisinthesmedirectionsthevector,ut ntimesslong. 8. Vectors of unit length Thereisonemorepieceofnottionweshllusewhenwritingvectors.If isnyvector,weshll write âtorepresentunitvectorinthedirectionof.auntvectorisvectorwhoselengthis 1,sotht â = 1. Thisnottiongivesusnotherwyofwritingthevector :wecnwriteits â,sothtitis the length multiplied y the unit vector â. Key Point Aunitvectorinthedirectionofthevector iswrittens â,sotht = â. 9. Using vectors in geometry Exmple There is useful theorem in geometry clled the midpoint theorem. In this theorem, we tke twopoints And B,definedwithrespecttonorigin O.Letuswrite forthepositionvector of A,nd forthepositionvectorof B.Wecnjoin And Bwithline,togivetringle. Nowtkethemidpoint Moftheline OA,ndthemidpoint Noftheline OB,ndjoin Mto Nwithline. Cnwesynythingoutthereltionshipetweentheline MN ndtheline AB? 6 c mthcentre 2009
7 M A O N B Wecnnswerthisveryesilywithvectors. Wecnwritethevectorforthelinesegment AB s AO + OB.Now AOisthereverseofthevector,soitis. And OBisthesmesthe vector. Therefore AB = AO + OB = ( ) + =. Whtout MN?Followingthesmeresoning,thisis MO +ON.Butwhtis MO?Thisis vectorhlfthelengthof AO,ndinthesmedirection,soitmuste 1 2 ( ).Inthesmewy, ONisinthesmedirections OB,utishlfthelength,soitmuste 1 2.Therefore MN = MO + ON = 1 2 ( ) = 1 ( ). 2 Nowwecncompre ABnd MN. Fromourclcultion,wecnseetht MNis 1 AB. So, 2 sthisisvectoreqution,ittellsustwothings. First,ittellsusoutmgnitude,sotht MN = 1 AB.Also,ittellsustht MNnd ABmusteinthesmedirection,sotht MN//AB. 2 Thisisclledthemidpointtheoremfortringle. Itsttesthtifyoujointhemidpointsof twosidesoftringlethentheresultinglineisequltohlfofthethirdsideofthetringle,nd isprlleltoit. Exmple Wecnpplythemidpointtheoremtoqudrilterl,orindeedtonyfourpointsinspce,to giveninterestinggeometriclresult. Weshllcllthefourpoints A, B, Cnd D. Weshll lsogivelelstothemidpointsofthefoursides:weshllcllthemidpoints P, Q, Rnd S. Nowletusjointhefourmidpoints,tomkenewshpe PQRS.Whtkindofshpeisthis? B Q C R P D A S 7 c mthcentre 2009
8 Wecnidentifytheshpeyjoiningthepoints And C. Ifwepplythemidpointtheoremtotringle ABC,weseetht PQ = 1 2 AC. Butifwepplythemidpointtheoremtothetringle ADC,welsoseetht RS = 1 2 AC. If we comine these two equtions, we then otin PQ = RS. Nowthisisvectoreqution,ndsoittellsustwothings. First,ittellsusthtthelengthof PQisthesmesthelengthof RS. Andsecondly,ittellsusthtthedirectionof PQisthe smesthedirectionof RS,sotht PQnd RSreprllel. Buthvingtwoprllelsidesof equllengthispropertywhichdefinesprllelogrm,ndsotheshpe PQRSmuste prllelogrm. Exmple Weshllnowusevectorstoproveonemoretheorem. Tketwopoints And B,hvingpositionvectors, withrespecttonorigin O. Drwthe line AB,ndtkepoint P onthtlinewhichdividesitinthertioof mto n. Whtisthe positionvectorof Pwithrespectto O?. A m P r n B O Wecnusethesmemethodthtweusedefore.Weknowtht ndwelsoknowtht OA =.Butwhtis AP? OP = OA + AP, (1) Now APisinthesmedirections AB,ndtheirlengthsreinthertioof mto m + n.so Welsoknowtht AP = m AB. (2) m + n AB = AO + OB =. 8 c mthcentre 2009
9 Now we cn put these three sttements together, replcing AP in eqution(1) y using eqution(2),ndreplcing ABineqution(2)yusingtheeqution(3),sothteverythingwille writtenintermsof nd.thisgivesus OP = + m ( ). m + n Putting ll this over common denomintor then gives OP = (m + n) + m( ). m + n Ifweexpndtherckets,theterm mwillcncelwiththeterm m( ),ndsofinllywehve OP = n + m m + n. Thisformulgivesuswyofclcultingthepositionvectorofthepoint P.Forinstnce,if m nd nwereoth 1then Pwouldethemidpointof AB. Thepositionvectorofthemidpoint woulde ( + )/2.Asnotherexmple,if m = 2nd n = 1,sotht Pwstwothirdsofthe wylongtheline,thenthepositionvectorof Pwoulde ( + 2)/3. Exercises 1.Thevector isshownelow. Sketchthevectors 2, 3, 1 nd In OAB, OA = nd OB =.Intermsof nd, () Wht is AB? () Wht is BA? (c) Whtis OP,where Pisthemidpointof AB? (d) Whtis AP? (e) Whtis BP? (f) Whtis OQ,where Qdivides ABinthertio2:3? 3.Whtismentyunitvector? 4.If eisunitvector,whtisthelengthof 3e? 5.In ABC, AB =, BC =, CA = c.whtis + + c? 9 c mthcentre 2009
10 Answers () () (c) 1 1 (e) ( ) (f) Avectorwithlength ( + )(d) 1( ) c mthcentre 2009
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