Introduction to vectors mcTYintrovector Avectorisaquantitythathasbothamagnitudeorsizeanda direction
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Introduction to vectors mcTYintrovector Avectorisaquantitythathasbothamagnitudeorsizeanda direction

Bothofthese propertiesmustbegiveninordertospecifyavectorcomplet elyInthisunitwedescribehowto writedownvectorshowtoaddandsubtractthemandhowtous ethemingeometry Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exerci

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Introduction to vectors mcTYintrovector Avectorisaquantitythathasbothamagnitudeorsizeanda direction




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Introduction to vectors mc-TY-introvector-2009-1 Avectorisaquantitythathasbothamagnitude(orsize)anda direction. Bothofthese propertiesmustbegiveninordertospecifyavectorcomplet ely.Inthisunitwedescribehowto writedownvectors,howtoaddandsubtractthem,andhowtous ethemingeometry. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: distinguishbetweenavectorandascalar; understandhowtoaddandsubtractvectors;

knowwhenonevectorisamultipleofanother; usevectorstosolvesimpleproblemsingeometry. Contents 1. Introduction 2 2. Representingvectorquantities 2 3. Positionvectors 3 4. Somenotationforvectors 3 5. Addingtwovectors 4 6. Subtractingtwovectors 5 7. Addingavectortoitself 5 8. Vectorsofunitlength 6 9. Usingvectorsingeometry 6 www.mathcentre.ac.uk 1 math centre2009
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1. Introduction Vectorquantitiesareextremelyusefulinphysics.Theimpo rtantcharacteristicofavectorquan- tityisthatithasbothamagnitude(orsize)andadirection. Bothofthesepropertiesmustbe giveninordertospecifyavectorcompletely.

Anexampleofavectorquantityisadisplacement.Thistellu showfarawaywearefromafixed point,anditalsotellsusourdirectionrelativetothatpoi nt. Anotherexampleofavectorquantityisvelocity. Thisisspe ed,inaparticulardirection. An exampleofvelocitymightbe60mphduenorth. Aquantitywithmagnitudealone,butnodirection,isnotave ctor. Itiscalleda scalar instead. Oneexampleofascalarisdistance.Thistellsushowfarwear efromafixedpoint,butdoesnot giveusanyinformationaboutthedirection.Anotherexampl eofascalarquantityisthemassof anobject. Key Point Avectorhasbothmagnitudeanddirection,andboththesepro

pertiesmustbegiveninorder tospecifyit.Aquantitywithmagnitudebutnodirectionisc alledascalar. 2. Representing vector quantities Wecanrepresentavectorbyalinesegment.Thisdiagramshow stwovectors. Wehaveusedasmallarrowtoindicatethatthefirstvectorisp ointingfrom to .Avector pointingfrom to wouldbegoingintheoppositedirection. Sometimeswerepresentavectorwithasmalllettersuchas ,inaboldtypeface. Thisis commonintextbooks,butitisinconvenientinhandwriting. Inwriting,wenormallyputabar underneath,orsometimesontopof,theletter: or .Inspeech,wecallthisthevector -bar. www.mathcentre.ac.uk 2 math centre2009


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3. Position vectors Sometimesvectorsarereferredtoafixedpoint,anorigin. Su chavectoriscalledaposition vector. Sowemightrefertothepositionvectorofapoint withrespecttoanorigin . In writing,mightput OP forthisvector.Alternatively,wecouldwriteitas .Thesetwoexpressions refertothesamevector. 4. Some notation for vectors Whatdoesitmeanif,fortwovectors, ?Thismeansfirstthatthelengthof equalsthe lengthof ,sothatthetwovectorshavethesamemagnitude.Butitalsom eansthat and areinthesamedirection.Howcanwewritethisdownmoresucc inctly?

Iftwovectorsareinthesamedirection,thentheyarepara llel.Wewritethisdownas // Forlength,ifwehaveavector AB ,wecanwriteitslengthas AB withoutthebar.Alternatively, wecanwriteitas AB .Thetwoverticallinesgiveusthemodulus,orsizeof,theve ctor.Ifwe haveavectorwrittenas ,wecanwriteitslengthaseither withtwoverticallines,oras inordinarytype(orwithoutthebar).Thisiswhyitisveryim portanttokeeptotheconvention thathasbeenadoptedinordertodistinguishbetweenavecto randitslength. Key Point Thelengthofavector AB iswrittenas AB or AB andthelengthofavector iswrittenas (inordinarytype,orwithoutthebar)oras

Iftwovectors and areparallel,wewrite // www.mathcentre.ac.uk 3 math centre2009
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5. Adding two vectors Oneofthethingswecandowithvectorsistoaddthemtogether . Weshallstartbyadding twovectorstogether. Oncewehavedonethat,wecanaddanynu mberofvectorstogetherby addingthefirsttwo,thenaddingtheresulttothethird,ands oon. Inordertoaddtwovectors,wethinkofthemasdisplacements . Wecarryoutthefirstdis- placement,andthenthesecond. Sotheseconddisplacementm uststartwherethefirstone finishes. a Thesumofthevectors, (orthe resultant ,asitissometimescalled)iswhatwegetwhen

wejoinupthetriangle.Thisiscalledthe trianglelaw foraddingvectors. Thereisanotherwayofaddingtwovectors. Insteadofmaking thesecondvectorstartwhere thefirstonefinishes,wemakethembothstartatthesameplace ,andcompleteaparallelogram. Thisiscalledthe parallelogramlaw foraddingvectors. Itgivesthesameresultasthetriangle law,becauseoneofthepropertiesofaparallelogramisthat oppositesidesareequalandinthe samedirection,sothat isrepeatedatthetopoftheparallelogram. a Key Point Wecanaddtwovectors and bymaking startwhere finishes, andcompletingthe triangle.Alternatively,wecanmake and

startatthesameplace,andtakethediagonalof theparallelogram. www.mathcentre.ac.uk 4 math centre2009
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6. Subtracting two vectors Whatis ?Wethinkofthisas + ( ,andthenweaskwhat mightmean.Thiswill beavectorequalinmagnitudeto ,butinthereversedirection. Nowwecansubtracttwovectors.Subtracting from willbethesameas adding to a Key Point means + ( 7. Adding a vector to itself Whathappenswhenyouaddavectortoitself,perhapsseveral times? Wewrite,forexample, = 3 Inthesameway,wewouldwrite ... {z copies www.mathcentre.ac.uk 5 math centre2009
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Key Point Avector

isinthesamedirectionasthevector ,but timesaslong. 8. Vectors of unit length Thereisonemorepieceofnotationweshallusewhenwritingv ectors.If isanyvector,weshall write torepresentaunitvectorinthedirectionof .Auntvectorisavectorwhoselengthis ,sothat = 1 Thisnotationgivesusanotherwayofwritingthevector :wecanwriteitas ,sothatitis thelength multipliedbytheunitvector Key Point Aunitvectorinthedirectionofthevector iswrittenas ,sothat 9. Using vectors in geometry Example Thereisausefultheoremingeometrycalledthe mid-pointtheorem . Inthistheorem,wetake twopoints and ,definedwithrespecttoanorigin

.Letuswrite forthepositionvector of ,and forthepositionvectorof .Wecanjoin and withaline,togiveatriangle. Nowtakethemid-point oftheline OA ,andthemid-point oftheline OB ,andjoin to withaline. Canwesayanythingabouttherelationshipbetwe entheline MN andtheline AB www.mathcentre.ac.uk 6 math centre2009
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Wecananswerthisveryeasilywithvectors. Wecanwritethev ectorforthelinesegment AB as AO OB .Now AO isthereverseofthevector ,soitis . And OB isthesameasthe vector .Therefore AB AO OB = ( ) + Whatabout MN ?Followingthesamereasoning,thisis MO ON .Butwhatis MO Thisisa vectorhalfthelengthof

AO ,andinthesamedirection,soitmustbe .Inthesameway, ON isinthesamedirectionas OB ,butishalfthelength,soitmustbe .Therefore MN MO ON ) + Nowwecancompare AB and MN . Fromourcalculation,wecanseethat MN is AB . So, asthisisavectorequation,ittellsustwothings. First,it tellsusaboutmagnitude,sothat MN AB .Also,ittellsusthat MN and AB mustbeinthesamedirection,sothat MN//AB Thisiscalledthemid-pointtheoremforatriangle. Itstate sthatifyoujointhemid-pointsof twosidesofatrianglethentheresultinglineisequaltohal fofthethirdsideofthetriangle,and isparalleltoit. Example

Wecanapplythemid-pointtheoremtoaquadrilateral,orind eedtoanyfourpointsinspace,to giveaninterestinggeometricalresult. Weshallcallthefo urpoints and . Weshall alsogivelabelstothemid-pointsofthefoursides:weshall callthemid-points and Nowletusjointhefourmid-points,tomakeanewshape PQRS .Whatkindofshapeisthis? www.mathcentre.ac.uk 7 math centre2009
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Wecanidentifytheshapebyjoiningthepoints and Ifweapplythemid-pointtheoremtotriangle ABC ,weseethat PQ AC. Butifweapplythemid-pointtheoremtothetriangle ADC ,wealsoseethat RS AC. Ifwecombinethesetwoequations,wethenobtain PQ RS.

Nowthisisavectorequation,andsoittellsustwothings. Fi rst,ittellsusthatthelengthof PQ isthesameasthelengthof RS . Andsecondly,ittellsusthatthedirectionof PQ isthe sameasthedirectionof RS ,sothat PQ and RS areparallel. Buthavingtwoparallelsidesof equallengthisapropertywhichdefinesaparallelogram,and sotheshape PQRS mustbea parallelogram. Example Weshallnowusevectorstoproveonemoretheorem. Taketwopoints and ,havingpositionvectors withrespecttoanorigin . Drawthe line AB ,andtakeapoint onthatlinewhichdividesitintheratioof to . Whatisthe positionvectorof withrespectto ?.

Wecanusethesamemethodthatweusedbefore.Weknowthat OP OA AP, (1) andwealsoknowthat OA .Butwhatis AP Now AP isinthesamedirectionas AB ,andtheirlengthsareintheratioof to .So AP AB. (2) Wealsoknowthat AB AO OB www.mathcentre.ac.uk 8 math centre2009
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Nowwecanputthesethreestatementstogether,replacing AP inequation(1)byusingequa- tion(2),andreplacing AB inequation(2)byusingtheequation(3),sothateverything willbe writtenintermsof and .Thisgivesus OP Puttingallthisoveracommondenominatorthengives OP Ifweexpandthebrackets,theterm willcancelwiththeterm ,andsofinallywehave OP

Thisformulagivesusawayofcalculatingthepositionvecto rofthepoint .Forinstance,if and wereboth then wouldbethemid-pointof AB . Thepositionvectorofthemidpoint wouldbe .Asanotherexample,if = 2 and = 1 ,sothat wastwo-thirdsofthe wayalongtheline,thenthepositionvectorof wouldbe + 2 Exercises 1.Thevector isshownbelow. Sketchthevectors and 2.In OAB OA and OB .Intermsof and (a) Whatis AB (b) Whatis BA (c) Whatis OP ,where isthemidpointof AB (d) Whatis AP (e) Whatis BP (f) Whatis OQ ,where divides AB intheratio2:3? 3.Whatismeantbyaunitvector? 4.If isaunitvector,whatisthelengthof 5.In ABC AB BC CA .Whatis

www.mathcentre.ac.uk 9 math centre2009
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Answers 1. 2. (a) (b) (c) (d) (e) (f) 3.Avectorwithlength1 4.3 5.0 www.mathcentre.ac.uk 10 math centre2009