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.)Afteragoodamountofpra