Introduction to vectors mc-ty-introvector-2009-1 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 6 www.mthcentre.c.uk 1 c mthcentre 2009
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. www.mthcentre.c.uk 2 c mthcentre 2009
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 // www.mthcentre.c.uk 3 c mthcentre 2009
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. www.mthcentre.c.uk 4 c mthcentre 2009
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 www.mthcentre.c.uk 5 c mthcentre 2009
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 mid-point theorem. In this theorem, we tke twopoints And B,definedwithrespecttonorigin O.Letuswrite forthepositionvector of A,nd forthepositionvectorof B.Wecnjoin And Bwithline,togivetringle. Nowtkethemid-point Moftheline OA,ndthemid-point Noftheline OB,ndjoin Mto Nwithline. Cnwesynythingoutthereltionshipetweentheline MN ndtheline AB? www.mthcentre.c.uk 6 c mthcentre 2009
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 = 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 Thisisclledthemid-pointtheoremfortringle. Itsttesthtifyoujointhemid-pointsof twosidesoftringlethentheresultinglineisequltohlfofthethirdsideofthetringle,nd isprlleltoit. Exmple Wecnpplythemid-pointtheoremtoqudrilterl,orindeedtonyfourpointsinspce,to giveninterestinggeometriclresult. Weshllcllthefourpoints A, B, Cnd D. Weshll lsogivelelstothemid-pointsofthefoursides:weshllcllthemid-points P, Q, Rnd S. Nowletusjointhefourmid-points,tomkenewshpe PQRS.Whtkindofshpeisthis? B Q C R P D A S www.mthcentre.c.uk 7 c mthcentre 2009
Wecnidentifytheshpeyjoiningthepoints And C. Ifwepplythemid-pointtheoremtotringle ABC,weseetht PQ = 1 2 AC. Butifwepplythemid-pointtheoremtothetringle 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 =. www.mthcentre.c.uk 8 c mthcentre 2009
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 Pwouldethemid-pointof AB. Thepositionvectorofthemidpoint woulde ( + )/2.Asnotherexmple,if m = 2nd n = 1,sotht Pwstwo-thirdsofthe wylongtheline,thenthepositionvectorof Pwoulde ( + 2)/3. Exercises 1.Thevector isshownelow. Sketchthevectors 2, 3, 1 nd 2. 2 2.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? www.mthcentre.c.uk 9 c mthcentre 2009
Answers 1. 2 1 2 3 2 2. () () (c) 1 1 (e) ( ) (f) 3 + 2 2 5 5 3.Avectorwithlength1 4.3 5.0 ( + )(d) 1( ) 2 2 www.mthcentre.c.uk 10 c mthcentre 2009