Abstract Vectors Vectors1Notes

GeneralTips: ● Drawingadiagramhelps,evenifyoudon’thavespecificcoordinates.● Rememberyourtools:○ RatioTheorem○ ComparingCoefficients○ Parallelifandonlyifka=b○ 𝑎 • 𝑏 = 𝑎| | 𝑏| |𝑐𝑜𝑠θ , 𝑎 × 𝑏 = 𝑎| | 𝑏| |𝑠𝑖𝑛θ (𝑛)○ Bereadytouseacombinationofthesetools. ● Rememberpropertiesofdot/crossproduct.( ,α × 𝑏 = − 𝑏 × 𝑎, 2α × 𝑏 = α × 2𝑏 distributiveness,etc.)● Remembertheangleformofyourdot/crossproducts.● Morespecificstuff: 0impliesperpendicularvectors, 0impliesparallel𝑎 • 𝑏 = 𝑎 × 𝑏 =vectors BasicTools: ,thesignofthedotproducttellsyouwhethertheanglebetween𝑎 • 𝑏 = |𝑎||𝑏|𝑐𝑜𝑠(Θ)thetwovectorsisacuteorobtuse.(Pointingroughlyinthesamedirectionornot). 𝑎 × 𝑏 = |𝑎||𝑏|𝑠𝑖𝑛(Θ)𝑛Whereistheunitvectorperpendiculartobothvectors,whosedirectionisgivenbythe𝑛righthandrule. Inparticular:If and areperpendicular,𝑎 𝑏 𝑎 • 𝑏 = 0If and areparallel,𝑎 𝑏 𝑎 × 𝑏 = 0givestheareaoftheparallelogramwhosesidesare and.(Divideby2toget|𝑎 × 𝑏| 𝑎 𝑏areaoftriangleOAB) Foranytwovectors ,theangle betweenthemisgivenby:𝑎,𝑏 Θ𝑐𝑜𝑠(Θ) = (𝑎•𝑏)|𝑎||𝑏| Lengthofprojectionof onto isgivenby:𝑎 𝑏 𝑎 • 𝑏|𝑏| = 𝑎 • 𝑏 Createdby:u/A_Strolling_Orca 1

RatioTheorem:GiventwopointsAandBwithpositionvectorsand,theposition𝑎 𝑏vectorofthepointC,thathasthepropertyAC:CB= ,isgivenby:λ:µ 𝑂𝐶= (λ𝑂𝐵+ µ𝑂𝐴)/(λ + µ) Thiscanalwaysberescaledsuchthat ,whichgives:λ + µ = 1𝑂𝐶= (1 − µ)𝑂𝐵+ µ𝑂𝐴 Twovectorsand areparallelifandonlyif:𝑎 𝑏,forsomenonzerorealnumber𝑎 = 𝑘𝑏 𝑘 (ComparingCoefficients)Fortwonon-parallelvectorsand:𝑎 𝑏λ𝑎 + µ𝑏 = 𝑢𝑎+ 𝑣𝑏 ↔ λ = 𝑢, µ = 𝑣Thisisusefulwhensolvingforintersectionsbetweentwolines. UsefulIdentities: ,andinparticular:|𝑎| 2 = 𝑎 • 𝑎|𝑎+ 𝑏| 2 = (𝑎 + 𝑏) • (𝑎 + 𝑏) = |𝑎| 2 + |𝑏| 2 + 2(𝑎 • 𝑏) (Analogoustodifferenceof2squares)|𝑎| 2 − |𝑏| 2 = (𝑎 − 𝑏) • (𝑎 + 𝑏) Overview: Thedifficultyinabstractvectorquestionsliesinidentifyingwhichformulaeareusefulforagivenquestion.Inappliedvectorquestions,mostquestionpartsaskyoutosolveforthingswhichareclearlyassociatedwithacertainformula/methodthatyouareexpectedtoknow.(E.g.Footofperpendicular,lineequation,etc.)Thevariousformulaerequiredforappliedvectorsaredifferentenoughtonotposeanyissuesinidentifyingwhichshouldbeused(atleast,aftersomepractice).Phraseddifferently,ifyouhaveallthevectorsformulaememorised,youshouldn’tfindanydifficultyinmostpartsofanappliedvectorsquestion. Createdby:u/A_Strolling_Orca 2

Incontrast,anabstractvectorsquestiontestsyourabilitytoidentifywhichformulaewillbeuseful.Whattheyrequireyoutosolvefor/proveoftenisn’tsomethingthatisinstantlygivenbyaformula.Theremayevenbemultipleplausiblewaystogoaboutsolvingaquestion,whichcancontributetothedifficultyofconsideringwhatformulatouse. Iwillwalkthroughmythoughtprocessforafewquestionsbelow,andhopefullyitwillshedsomelightonhowtosolvethesetypesofquestionsingeneral. GeneralPrinciples: - Almostalways,everypieceofinformationinthequestionwillbeuseful,andismeanttoinformyouofwhatformulaetoemploy.Certainphrases/keywordscanbeidentifiedandreliablyassociatedwithacertainformulatobeused.(Note:ThisappliestoalltopicsinH2math,notjustvectors.)Afteragoodamountofpractice,youcanevenforeseewhatquestionswillbeaskedaftersimplyreadingtheinformationgiven. - Knowingwhatquantitiesaformularelatestogetherwillhelpyouidentifywhichwillbeuseful.(E.g.knowingthevaluesof , ,andtheanglebetween|𝑎| 𝑎 • 𝑏 𝑎and,Icancalculate).Soifaquestionasksforananglebetweenvectors,𝑏 |𝑏|yououghttothinkoftheformulaethatcontainanglesbetweenvectors. - Thismaysoundobvious,buthavingyourformulaeatyourfingertips,oratleastwrittendownonroughpaper,willhelpgreatly. - Drawingadiagram,evenwithoutspecificcoordinates,canonlyhelpyou. Createdby:u/A_Strolling_Orca 3

Note:Isuggesttryingouteachquestionfirst,andtousepenandpapertofollowthesolutions. Examples: Example:RI2020PrelimP2 Iwillwalkthroughmythoughtprocessindoingthisquestion: i)CliesonOA,andDliesonOB,with and .Drawinga𝑂𝐶:𝐶𝐴= 2:1 𝑂𝐷:𝐵𝐷= 3:2diagram: Thus,itshouldbeclearthat ,and .𝑂𝐶= 2𝑎/3 𝑂𝐷= 3𝑏 Createdby:u/A_Strolling_Orca 4

ii)SincethequestionmentionedthelinesBCandandAD,wemightaswellfindtheirequations: ,and .𝐵𝐶: 𝑟 = 𝑏 + λ(𝑐 − 𝑏) 𝐴𝐷: 𝑟 = 𝑎 + µ(𝑑 − 𝑎) Wewereaskedtofindtheirintersection,soweshouldequatethesetwoequationswejustconstructed: .𝑎 + µ(𝑑 − 𝑎) = 𝑏 + λ(𝑐 − 𝑏) Additionally,we’remeanttoshowthepositionvectorof intermsof and,sowe𝑂𝐸 𝑎 𝑏shouldremoveand fromourequationbysubstitutingtheirvaluesinfrom5i).𝑐 𝑑 .𝑎 + µ(3𝑏 − 𝑎) = 𝑏 + λ(2𝑎/3− 𝑏) Thisseemsclosertowhatwewant.Now,theonlythingleftthatwecando(andhopefully,themostnaturalthingtodo,)isexpandthebrackets. (1 − µ)𝑎 + 3µ𝑏 = (2λ/3)𝑎 + (1 − λ)𝑏) Uponseeingthis,itshouldbenaturaltoequate/comparecoefficients.Ifitisn’tnatural,don’tworry.Itwillbecomenaturalwithpractice. Thus,weget: and .1 − µ = 2λ/3 3µ = 1 − λWhichwecanpunchG.C.tosolveandget: .µ =− 1/3,λ = 2 SincewewereaskedtofindOE,wesubthesebackintoget: ,asrequired.𝑂𝐸= (4/3)𝑎 − 1𝑏 iii)ThequestionwantsustoshowtheareaoftriangleCDEisamultipleof ,so𝑘|𝑎 × 𝑏|weoughttofirstwritedownanequationfortheareaofthetriangle.Andsinceweaimtoshowit’sacrossproduct,weshouldatleaststartbywritingitintermsofacrossproduct. AreaofTriangleCDE=(1/2)|𝐶𝐷× 𝐶𝐸| = (1/2)|(𝑂𝐷− 𝑂𝐶) × (𝑂𝐸− 𝑂𝐶)| = |(3𝑏 − (2/3)𝑎) × ((2/3)𝑎 − 𝑏)| Wecouldhavetakenadifferentpairofsides,buttheywillallgivethesameanswer. Createdby:u/A_Strolling_Orca 5

Now,sincetheendgoalwe’reaimingforhastheformk|axb|,weshouldexpandourbrackets. (1/2)|(3𝑏 − (2/3)𝑎) × ((2/3)𝑎 − 𝑏)| = (1/2)|2(𝑏 × 𝑎) + (2/3)𝑎 × 𝑏|= (1/2)|(− 4/3)𝑎 × 𝑏| Whereweutilisedthefactthat and𝑎 × 𝑎 = 0 𝑎 × 𝑏 =− 𝑏 × 𝑎 Now,pullingoutconstants, (1/2)|𝐶𝐷× 𝐶𝐸| = (2/3)|𝑎 × 𝑏| Andwegetouranswer, .𝑘 = 2/3 5iv)Thusfar,thequestionshavebeenquitestandardasfarasabstractvectorsquestionsgo.Likemanyotherquestions,thelastpartisthehardest,butcanbedonesmoothlyifyouknowwhattoconsider. WearegiventhatFliesonOBproduced,andOEbisectstheangleAOF,andweareaskedtofindtheratioOA:OB.Thisisequivalenttofinding .Recallthatbisecting|𝑎|/|𝑏|meanssplittinginto2equalparts,soangleAOE=angleEOF.Ingeneralwhendealingwithangles,thedotproductformula/formulaforanglebetweentwovectorscomesinuseful,soitmakessensetowritethisinformationintermsofit. 𝑐𝑜𝑠(𝐴𝑂𝐸) = 𝑂𝐴• 𝑂𝐸/|𝑂𝐴||𝑂𝐸|𝑐𝑜𝑠(𝐸𝑂𝐹) = 𝑂𝐸• 𝑂𝐹/|𝑂𝐸||𝑂𝐹| Sincetheanglesareequal,wemayequatethetwo: (𝑂𝐴• 𝑂𝐸)/|𝑂𝐴||𝑂𝐸| = (𝑂𝐸• 𝑂𝐹)/|𝑂𝐸||𝑂𝐹| Noticethattermscancelout: (𝑂𝐴• 𝑂𝐸)/|𝑂𝐴| = (𝑂𝐸• 𝑂𝐹)/|𝑂𝐹| Sinceweneedtogetthingsintermsof and ,itmakessensetowritetheabove|𝑎| |𝑏|lineintermsofthem.However,wefaceanissuewhendealingwithOF,asweonlyknowthatitliesonOBproduced.Thus,weknow forsomepositiveconstant,𝑂𝐹=− λ𝑏asitgoesintheoppositedirectionofOB. Createdby:u/A_Strolling_Orca 6

But,sincewedivideby|OF|,wehave: ,andwesee𝑂𝐹/|𝑂𝐹| =− λ𝑏/|− λ𝑏| =− 𝑏/|𝑏|thattheconstantcancelsoutonthetopandbottom.Thus,wecanproceedwithwritingtheequalityintermsof and :|𝑎| |𝑏| 𝑎 • ((4/3)𝑎 − 𝑏)/|𝑎| = ((4/3)𝑎 − 𝑏)) • (− 𝑏/|𝑏|)Expanding:(4/3)|𝑎| − (𝑎 • 𝑏)/|𝑎| = (− 4/3)(𝑎 • 𝑏)/|𝑏| + |𝑏| Weseemtobestuckagain,butagoodruleofthumbwhenyou’veusedalltheinformationgiventoyouistocheckwhatformulasyoucaneasilyapplytowhateveryou’veworkedupto.Inthiscase,wehaveyettoexpand ,andthiscangiveus(𝑎 • 𝑏)moretermsin and .|𝑎| |𝑏| (4/3)|𝑎| − |𝑏|𝑐𝑜𝑠(𝐴𝑂𝐵) = (− 4/3)|𝑎|𝑐𝑜𝑠(𝐴𝑂𝐵) + |𝑏| Sincewewant ,wecantrydividingby :|𝑎|/|𝑏| |𝑏|(4/3)|𝑎|/|𝑏| − 𝑐𝑜𝑠(𝐴𝑂𝐵) = (− 4/3)|𝑎|/|𝑏|𝑐𝑜𝑠(𝐴𝑂𝐵) + 1 Nowifwemakeitthesubject:|𝑎|/|𝑏| × (4/3)(1 + 𝑐𝑜𝑠(𝐴𝑂𝐵)) = (1 + 𝑐𝑜𝑠(𝐴𝑂𝐵)) Thus:|𝑎|/|𝑏| = 3/4 Andwearedone.(Thereisaone-linersolutionthatinvolvesgeometry,butthissortofthoughtprocessismorereliable.) Example:ASRJC2022PrelimP2 Createdby:u/A_Strolling_Orca 7

2i)WeneedtoshowA,B,Carecollinear.ThisrequiresustoshowthatanypairofAB,BC,orACarerelatedbyscalarmultiplication.Thatis,wecanmultiplyanyonebyascalartogetanother.(Notethatitissufficienttoshowthisholdsforanypairofthem.IchoseABandACsincetheypointinthesamedirection,whichshouldmakeformarginallyeasieralgebra.) 𝐴𝐵= 𝑏 − 𝑎 ,wherethissubstitutionfor isgiveninthequestion.𝐴𝐶= 𝑐 − 𝑎 = (λ − 1)𝑎 + µ𝑏 𝑐 Additionally,wehave .λ + µ = 1Thus:𝐴𝐶=− µ𝑎 + µ𝑏 = µ(𝑏 − 𝑎) = µ𝐴𝐵.Andthustheyarecollinear,asrequired. Aquickermethod:Noticethat isexactlythevectorthatdividesABintotheratio ,𝑂𝐶 λ:µbytheratiotheorem.AndifitdividesthelinesegmentAB,itmustlieonAB.HenceA,B,Carecollinear. 2ii)Itm