13vectors A-level考试纯数学P1课件.ppt

vectors

Translationsofaplane+cc0

SuchtranslationiscalledavectorandiswrittenlkHowwouldthetranslationofy=f(x)describedabovebeperformed?

+cc0

Vectoralgebraklsksl-k-lAnumbersuchassisoftencalledascalar

pqp+qp+qklmnp+q=q+pthisiscalledthecommutativeruleforadditionofvectors

ExampleIfandshowthatthereisanumberssuchthatp+sq=r

Theideaofadditioncanbeextendedtothreeormorevectorsqpr(p+q)+r=p+(q+r)Thisiscalledtheassociativeruleforadditionofvectors

Tocompletethealgebraofvectoraddition,thesymbol0isneededforthezerovectorDistributiverulesforvectors

Subtractionofvectorsqpx=q–pq+(-p)-p

BasicunitvectorsThevectorsandarecalledbasicunitvectorsp=ki+ljThevectorsofkiandljarecalledcomponentvectorsofpinthex-andy-directions

PositionvectorsVectorswrittenlikethisaresometimescalleddisplacementvectorsThetranslationfromOtoAiscalledthepositionvectorofAEGHFOAThepositionvectorofthepointAwithcoordinates(u,v)isOAa

AlgebrawithpositionvectorsPointsAandBhavepositionvectorsaandb.Findthepositionvectorsof(a)themid-pointMofAB(b)thepointoftrisectionTsuchthatExampleAMBABT

ExampleIntriangleABCthemid-pointsofBC,CAandABareD,EandF.ProvethatthelinesAD,BEandCF(calledthemedians)meetatapointG,whichisapointoftrisectionofeachofthemediansAGFEDCB.O

VectorsinthreedimensionsxyzOThepositionofapointisgivenbyitsthreecoordinates(x,y,z)pli+mj+nk

Afewdifferencesbetweentwodimensionsandthreedimensions.Theaxescanbetakeninpairstodefinecoordinateplanes.Forexample,thex-andy-axesdefinethehorizontalplane,calledthexy-plane.Allpointsinthisplanehavez-coordinatezero,sotheequationoftheplaneisz=0.similarlythexz-planeandtheyz-planehaveequatio