what is geometric algebra? - oulun lyseon lukio · 2016-01-29 · 5 proceed to the standard vector...
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CONTENTS0.ALKUSANAT/FOREWORD..............................................................................................................3
1.INTRODUCTION..............................................................................................................................4
2.THEALGEBRAOFREALNUMBERS(R)............................................................................................6
3.VECTORALGEBRA...........................................................................................................................8
4.SCALARPRODUCTORDOTPRODUCTa ∙ b..................................................................................12
5.COMPLEXPRODUCTa*binR2....................................................................................................16
6.VECTORPRODUCTORCROSSPRODUCTa×b..............................................................................25
7.QUATERNIONPRODUCTa ⋄ bINR4............................................................................................29
8.OUTERPRODUCTORWEDGEPRODUCTa ∧ b.............................................................................38
9.GEOMETRICPRODUCTANDGEOMETRICALGEBRA.....................................................................48
10.SCALARSANDPSEUDOSCALARS.................................................................................................67
11.FINALREMARKS..........................................................................................................................69
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0.ALKUSANAT/FOREWORDTämätekstionkoostelukuvuonna2015-2016esittämistäniviikkotarinoistaOulunLyseonlukion
matematiikankerhossa(lempinimeltäänGalois-kerho1).Esitintarinatsuomeksi,muttatekstion
englanninkielinenajatellenmahdollisiatuleviakinkerholaisia,joidensuomenkielentaitoon
puutteellinen.
LisäämälläotsikkoonmerkinnänPart1(Osa1)varaudunmahdollisuuteenjatkaageometrisen
algebrankäsittelyäpidemmällekin.
Tekstiiniliittyvätkommentitovattervetulleitakansisivullailmoitettuunsähköpostiosoitteeseen.
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ThesenotessummarizetheseriesoftalksIgaveinthemathclub(nicknamedGalois1WEWclub)of
OulunLyseo(aseniorhighschoolinOulu,Finland).
Theword”Part1”attheendofthetitleindicatesthatImightdecidetogoondeeperintothe
secretsofgeometricalgebra.
Iwishallremarksandcommentswelcometomyemail-addressgivenonthetitlepage.
Oulu,Finland
January2016
TeuvoLaurinolli
1ÉvaristeGalois(1811-1832),Frenchmathematicianwhosediscoveriesopenedthewaytoacomprehensivetheory–nowknownasGaloistheory–ofpolynomialequations.Hewasbadlywoundedinaduellanddiedattheageof20.
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1.INTRODUCTIONGeometryandalgebraarethetwomajorbranchesofmathematics.Geometry,themostancient
part2ofmathematics,considers2-dimensionalplanefigures(points,lines,triangles,
parallelograms,circlesetc)or3-dimensionalspatialobjects(planes,cubes,spheresetc)andtheir
properties.Algebra,originallysincetheMiddleAges3,hasbeenunderstoodasthestudyof
numbers4andtheiroperations(addition,multiplicationetc)usingsymbolicexpressions
(variables,polynomials,equationsetc).Acentralobjectiveofalgebraistoformulategeneral
rulesobeyedbytheseoperations.
Thesetwodomainsare,ofcourse,closelyintertwined.Weusealgebraicmethods(like
equations)tosolvegeometricproblemsandweusegeometrytoillustratealgebraicconcepts
(likeinterpretingtheproduct𝑎𝑏astheareaoftherectanglewithsides𝑎and𝑏).Thereisarich
varietyofinteraction.Avividexampleofthiswastheinventionofvectoralgebrainthe19th
century.Traditionallyalgebrahadbeenaboutsymboliccalculationswithnumbersbut
mathematicians(eg.Hamilton,GrassmannandClifford5)understoodthatitispossibleto
constructsystemsofalgebraicoperationsforgeometricobjects.Hamiltondidthisforvectors
(directedlinesegments).GrassmannandCliffordgeneralizedittoevenmorecomplexgeometric
objects.
Inthefollowingpageswewillmakeanoverviewoftheseideas.Westartbyreviewingthe
familiaralgebraofrealnumberswhichisthebasicmodelforalllaterextensions.Thenwe
2TheoldestknownmathematicalbookStoikheia(Elements)wasauthoredbytheGreekscholarEuclidca.300BC.Thebookpresentsalogicaldevelopmentofgeometryanditwascommonlyusedasahighschooltextbookuntil20thcentury.3ThePersianscholarMuḥammadibnMūsāal-Khwārizmī(ca.780–850AC)isrecognizedasthefatherofalgebra.Theword”algebra”wasaLatinizedversionoftheArabicwordal-jabr(meaning”collectingterms”)whichappearedinthenameofhisbook.4Thestudyofnumbers(natural,integer,rationaletc)andtheiroperations(addition,multiplicationetc)atthebasiclevel(withoutusinganylettersymbolslike𝑎, 𝑏, 𝑥, 𝑦, …)isusuallycalledarithmetic.5WilliamRowanHamilton(1805-1865),Irishmathematicianandphysicist;HermannGrassmann(1809-1877),Germanmathematicianandlinguist;WilliamKingdonClifford(1845-1879),Englishmathematician.
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proceedtothestandardvectoralgebraintwoandthreedimensions(planeandspace)aswellas
higherdimensions.Whiletheoperationofadditionhasanaturalandsimpledefinitionforvectors
thereareanumberofdifferentoptionsfortheoperationofmultiplication(product)ofvectors.
Theseincludescalar,complex,vector,quaternionic,outerandgeometricproducts.Wewill
considereachoftheseoptions–theirtechnicalpropertiesaswellasadvantagesand
disadvantages.OurfinalunifyingextensionisthegeometricalgebraalreadyinventedbyClifford
butforgottenforacenturyuntilrecentlywhenitwasfoundausefultoolincomputergame
programmingandphysics.
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2.THEALGEBRAOFREALNUMBERS(ℝ)Thesetℝofrealnumberscontainsallintegers(0,±1,±2,±3,…)aswellasrationals(quotients
𝑚/𝑛,where𝑚and𝑛areintegersand𝑛 ≠ 0)andirrationals(like 2, 𝜋, …).Eachrealnumber𝑥
isrepresentedbyapointonthenumberline.Theabsolutevalue 𝑥 equalsthedistanceofthe
point𝑥fromtheoriginwhichitselfcorrespondsto0asshowninFigure1below.
Figure1.Thenumberline.Theorigin0isrepresentedbytheredpoint.Therealnumbers𝑥 ≈ 2.6and𝑦 ≈−1.7arerepresentedbythebluepoints.Their(positive)absolutevalues 𝑥 and 𝑦 indicatethedistancesfromtheorigintothesepoints.
Theusefulnessofnumbersislargelyduetothefactthatwecancalculate(andthatiswhat
algebraisabout)withthem.Wecanadd,subtract,multiply,divide,raisetopowers,takesquare
rootsetc.Amongthesethemostfundamentalarebinaryoperations6ofadditionand
multiplication.Whenappliedtotwonumbers𝑥and𝑦theadditionproducestheirsum𝑥 + 𝑦and
multiplicationproducestheirproduct𝑥𝑦(alsodenotedby𝑥 ∙ 𝑦or𝑥×𝑦).Otheroperationscanbe
logicallyreducedtothesetwofundamentaloperations7onwhichwethereforeconcentratein
whatfollows.Inthesenotesthewordalgebraissynonymoustothestudyofadditionand
multiplicationandtheircalculationrules.
Inthealgebraofℝwehavethefollowingfamiliarruleswhichholdtrueforallrealnumbers
𝑎, 𝑏, 𝑐.
6Additionisbinaryinthesensethatittakestwonumbers(e.g.5and7)andreturnsonenumber(theirsum12).Likewisemultiplication.7Forexample,thedifference𝑥 − 𝑦isdefinedasthesum𝑥 + (−𝑦),andthequotient𝑥/𝑦astheproduct𝑥 ∙ 1/𝑦 .Here – 𝑦 isthenegativeof𝑦and 1/𝑦 istheinverseof𝑦.
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𝑎 + 𝑏 = 𝑏 + 𝑎 (additioniscommutative)
(𝑎 + 𝑏) = 𝑎 + (𝑏 + 𝑐) (additionisassociative)
𝑎𝑏 = 𝑏𝑎 (multiplicationiscommutative)
𝑎𝑏 𝑐 = 𝑎(𝑏𝑐) (multiplicationisassociative)
𝑐(𝑎 + 𝑏) = 𝑐𝑎 + 𝑐𝑏 (multiplicationisdistributiveoveraddition)
Noticethatcommutativityandassociativityarepropertiesofonesingleoperationonlywhile
distributivitytiesthetwooperationstogether.Noticealsothatassociativityrulesallowustowrite
”triple”expressionslike𝑎 + 𝑏 + 𝑐and𝑎𝑏𝑐becauseitdoesnotmatterhowwechoosetopairthe
numberswhencalculatingtheresult.Thesameistrueforanyfinitesum𝑎G + 𝑎H + ⋯+ 𝑎Jor
product𝑎G𝑎H …𝑎J.
Inthesenoteswewillfocusontheabovementionedrulesofcommutativity,associativityand
distributivityalthoughthereareotherimportantrulesthatadditionandmultiplicationsatisfy8.
Inthenextchapterswewillusethealgebraofrealnumbersasamodelforsettingupsimilar
systemsforgeometricobjectslikevectors.
8Theserulesincludetheexistenceofsocalledneutralandinverseelements.Foradditiontheneutralelementis0andtheinverseof𝑥is−𝑥whichexistsforallreals𝑥.Formultiplicationtheneutralelementis1andtheinverseof𝑥is1/𝑥whichexistsforallnonzeroreals𝑥.
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3.VECTORALGEBRAWebeginbyconsideringvectorsinthetwo-dimensionalplaneℝH.(Thesymbolℝdenotesthe
one-dimensionallinewherethelocationofeachpointcanbespecifiedbyonerealnumber.Inthe
planeyoucanspecifyeachpointbytworealnumbers,forexamplebythe𝑥-and𝑦-coordinates9of
thepoint.HencethesymbolℝHfortheplane.)
VectorsinℝH(orplanevectors)representdisplacementsintheplane.Tospecifyadisplacement
(i.e.avector)youhavetospecifyitsdirectionandlength(alsocalledabsolutevalueofthevector).
Allvectorswiththesamedirectionandlengthareconsideredequal(seeFigure2).
Figure2.DisplacementsABandCDareequal,sobothrepresentthesamevector𝒃.SimilarlyAEandFGarethesamevector𝒂.But𝒂 ≠ 𝒃becausetheyhavedifferentdirections(eveniftheyhaveequallengths).Wefollowheretheusualconventionofdenotingvectorswithboldfacesymbols𝒂, 𝒃, …indistinctionfromrealnumbers(orscalars)𝑎, 𝑏, ….
Vectorshaveturnedoutveryusefultoolsinmanyareasandparticularlyinphysics.Greatmany
physicalmagnitudeshaveavectorcharacter.Familiarexamplesinmechanicsareposition,
velocity,accelerationandforce.Instudyingthemotionofaship,forexample,itiscrucialtoknow
notonlytheabsolutevalue(magnitude)butalsothedirectionofitsvelocity.Incontrasttovector
magnitudesinphysicsthereareotherdirectionlessmagnitudesliketime,mass,temperatureetc.
9Coordinatescanbedefinedbyplacingtwonumberlinesperpendiculartoeachothersothattheiroriginscoincide.Calltheselines𝑥-and𝑦-axes.ThecoordinatesofanypointPintheplanearethenobtainedasorthogonal(perpendicular)projectionsofPontheseaxes.
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Theyarecalledscalar10magnitudes.Theyaredefinedbyasinglerealnumberandaunitlike,for
example42.3seconds,56kilogramsetc.
Howthencanwedoalgebrawithvectors?Forthatpurposeweneedtodefinethebinary
operationsofaddition(sumoftwovectors)andmultiplication(productoftwovectors)ina
reasonableway.
ADDITIONOFVECTORS
Todefinethesumofvectors𝒂and𝒃itisusefultothinkthemasdisplacements.Thetotal
displacement(firstby𝒂andthencontinuedby𝒃)wouldbeanaturalcandidatefor𝒂 + 𝒃.Andin
factthisdefinitionprovidesuswithanotionofsumwhichenjoysallthenicepropertiesthathold
trueforthesumsofrealnumbers(seeFigure3).
Figure3.Thedefinitionofvectoradditionasthetotaldisplacement.Thesumoftwovectorsisclearlycommutative.Wecansimilarlydefinethesumofthreeofmorevectorsandthissumisalsocommutative(independentoftheorder)andassociative(independentofpairing).
SCALARMULTIPLICATIONOFVECTORS
Beforetryingtodefinetheproductoftwovectorsitisusefultointroducetheoperationofscalar
multiplicationwherewemultiplyavectorbyarealnumber(scalar).Thisoperationmaybeseen
simplyasanotherwayofdenotingsumswherethesamevectorisrepeated.Sowewrite
10Inphysicsrealnumbersareoftencalledscalarsbecausetheyexpressaquantity(likemass)inscale,thatis,inrelationtothechosenunit(likekilogram).Thesameterminologyisusedgenerallyinvectoralgebra.Theabsolutevalue(magnitudeorlength)ofavectorisgivenbyascalar.Buttodefineaplanevector(vectorinℝH)completelywehavetospecifyalsoitsdirectionbyanotherscalar(theanglethevectormakeswithagivenaxis).Foravectorinthe3-dimensionalspaceℝM,specifyingitsdirectionrequirestwoscalars,i.e.threescalarsaltogetherasoneisrequiredfortheabsolutevalue.Thisgeneralizestohigherdimensions:youneed𝑛scalarstospecifyavectorinℝJ.
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𝒂 + 𝒂 = 2𝒂,𝒂 + 𝒂 + 𝒂 = 3𝒂,etc
Soforapositiveinteger𝑛thevector𝑛𝒂hasthesamedirectionas𝒂butis𝑛timesaslongas𝒂.
Thisideaworksnaturallyforageneralscalarmultiple𝑥𝒂where𝑥isanyrealnumber(not
necessarilyaninteger).Incase𝑥isnegativethedirectionof𝒂isreversed.
Withsumsandscalarmultiplesdefinedwecanmeaningfullywritealgebraicvectorexpressions
like2𝒂 + 5𝒃,GM𝒂 − 𝒃 2,𝑘(𝒂 + 𝑙𝒃),etc.
ORTHONORMALBASISVECTORS
TheplaneℝHinwhichourvectors”live”isoftenequippedwithasystemofcoordinatesconsisting
oftwoorthogonalnumberlines(𝑥-and𝑦-axes)withthecommonorigin.Itisthenusefulto
introducebasisvectors𝒆Gand𝒆Hwhichareparallelwiththeaxesandhaveunitlength.
Figure4.Orthonormalbasis 𝒆G, 𝒆H fortheplaneℝHconsistsoftwoperpendicularunitvectors𝑒Gand𝑒Halignedwiththecoordinateaxes.Everyvectorintheplanecanbeexpressedasalinearcombinationofthebasisvectors.Forexample,𝒂 = 2𝒆G + 2𝒆Hand𝒃 = −𝒆G + 2𝒆H.Generally,if𝒓isapositionvector
11ofthepoint(𝑥, 𝑦)then𝒓 = 𝑥𝒆G + 𝑦𝒆H.
Ifvectorsaregivenintermsoftheorthonormalbasiswecanaddandscalarmultiplythemexactly
aswedowithapples(𝒆G)andoranges(𝒆H).Forexample,inFigure4wehave
11ThepositionvectorofagivenpointPextendsfromtheorigintoP.InFigure4,forexample,thevector𝒂isthepositionvectorofthepointP=(2,2).
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𝒂 + 𝒃 = (2 − 1)𝒆G + 2 + 2 𝒆H = 𝒆G + 4𝒆H
3𝒃 = 3(−𝒆G) + 3 ∙ 2𝒆H = −3𝒆G + 6𝒆H
Soyouaddvectorssimplybycollectingapplesandorangesandyouscalarmultiplyavectorsimply
bymultiplyingbothapplesandorangesbythisscalar.
REMARKONBASES
Infact,anytwonon-parallelvectorscanbeusedasabasisfortheplaneℝHbutitismost
convenienttochoosetheorthonormalbasis 𝒆G, 𝒆H whichhasanaturalconnectiontothe
coordinatesasindicatedinthesubtextofFigure4.Oneniceconsequenceofthisconnectionis
thatwecaneasilycalculatetheabsolutevalues(lengths)anddirectionanglesofvectorsspecified
inthisbasis.Take,forexample,vector𝒃 = −1𝒆G + 2𝒆HinFigure4.ByPythagoras’Theoremwe
have 𝒃 = (−1)H + 2H = 5.(Forageneralvector𝒓 = 𝑥𝒆G + 𝑦𝒆Hwehaveananalogous
expressionforitsabsolutevalue, 𝒓 = 𝑥H + 𝑦H.)Thedirectionof𝒃canbeexpressedbythe
angle𝛽itmakeswiththe𝑥-axis.Simplecirculartrigonometrygivestan 𝛽 = 2 (−1) = −2from
whichcalculator(andabitofreflection)gives𝛽 ≈ 116°.(Ingeneralthedirectionangle𝜃ofthe
vector𝒓 = 𝑥𝒆G + 𝑦𝒆Hsatisfiestan 𝜃 = 𝑦 𝑥.)
Wehavesuccessfullycompletedthedefinitionofvectoradditionandscalarmultiplicationwhichis
aspecialformofaddition.Theseoperationsremainunchangedwhenwegoovertohigher
dimensionsℝM,ℝY,etcandallfamiliarrules(commutativityandassociativityinparticular)remain
trueforthem.
Wenowturnourattentiontoaharder(andmoreinteresting)question:howshouldwedefinethe
multiplicationoperationforvectorsinℝHand(later)inhigherdimensions?Mathematicianshave
foundmanyanswerstothisquestion.Inthenextchapterswewillconsiderthefollowingvariants
fortheproductoftwovectors𝒂and𝒃:(1)scalarproduct/dotproduct/innerproduct𝒂 ∙ 𝒃,(2)
complexproduct𝒂 ∗ 𝒃,(3)vectorproduct/crossproduct𝒂×𝒃,(4)quaternionproduct𝒂 ⋄ 𝒃,(5)
outerproduct/exteriorproduct𝒂⋀𝒃andfinally(6)geometricproduct𝒂𝒃.
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4.SCALARPRODUCTORDOTPRODUCT𝒂 ∙ 𝒃
Thisversionofvectormultiplication,denotedby𝒂 ∙ 𝒃(hencethenicknamedotproduct),was
originallyinventedforphysicalapplicationsandparticularlyforthecalculationofphysicalwork
(𝑊,ascalarvalue)donebyaforce(vector𝑭)draggingaloadoveradistance(adisplacement𝒔,
anothervector).Itturnsoutthat𝑊 = 𝑭𝒔 ∙ 𝒔 where𝑭𝒔istheprojectionof𝑭alongthelineof
displacementvector𝒔.Theexpression 𝑭𝒔 ∙ 𝒔 wasthendefinedtobe𝑭 ∙ 𝒔,thescalarproductof
thevectors𝑭and𝒔.ThisideaisgeneralizedforallvectorsinFigures5and6below.
Figure5.Scalarproductofvectors𝒂and𝒃incasetheymakeanacuteangle(0° ≤ 𝜃 ≤ 90°).LEFT:vector𝒃𝒂istheprojectionof𝒃onthelineofvector𝒂.RIGHT:vector𝒂𝒃istheprojectionof𝒂onthelineofvector𝒃.FromthesimilartrianglesOBCandOADitfollowsthattheproductsoflengths 𝒂 ∙ 𝒃𝒂 and 𝒃 ∙ 𝒂𝒃 areequal.Thisproductiscalledthescalarproduct𝒂 ∙ 𝒃ofthevectors𝒂and𝒃.Hencewehave𝒂 ∙ 𝒃 = 𝒂 ∙ 𝒃𝒂 = 𝒃 ∙ 𝒂𝒃 .Itiscalledscalarproductbecausetheanswerisa(non-negative)realnumber.
Figure6.Scalarproductofvectors𝒂and𝒃incasetheymakeanobtuseangle(90° < 𝜃 ≤ 180°).LEFT:vector𝒃𝒂istheprojectionof𝒃onthelineofvector𝒂.Now𝒃𝒂and𝒂haveoppositedirections.RIGHT:vector𝒂𝒃istheprojectionof𝒂onthelineofvector𝒃.Now𝒂𝒃and𝒃haveoppositedirections.FromthesimilartrianglesOBCandOADitfollowsthattheproductsoflengths 𝒂 ∙ 𝒃𝒂 and 𝒃 ∙ 𝒂𝒃 areequal.Nowthenegativeofthisproductiscalledthescalarproduct𝒂 ∙ 𝒃ofthevectors𝒂and𝒃.Hencewehave𝒂 ∙ 𝒃 = − 𝒂 ∙ 𝒃𝒂 = − 𝒃 ∙ 𝒂𝒃 .Itiscalledscalarproductbecausetheanswerisa(negative)realnumber.
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Cosineformulaforscalarproduct
Thedefinitionpresentedinfigures5and6canbesummarizedbyasimpleformula:
𝒂 ∙ 𝒃 = 𝒂 ∙ 𝒃 ∙ cos 𝜃
whichistruebecause 𝒃 ∙ cos 𝜃 = + 𝒃𝒂 if0° ≤ 𝜃 ≤ 90°− 𝒃𝒂 if90° < 𝜃 ≤ 180°.
Consequences
(1) 𝒂 ∙ 𝒂 = 𝒂 H
(2) 𝒂 ∙ 𝑡𝒂 = 𝑡 𝒂 H[𝑡isascalar]
(3a) If𝒂and𝒃havethesamedirection(𝜃 = 0°),then𝒂 ∙ 𝒃 = 𝒂 ∙ 𝒃
(3b) If𝒂and𝒃havetheoppositedirection(𝜃 = 180°),then𝒂 ∙ 𝒃 = − 𝒂 ∙ 𝒃
(4) If𝒂and𝒃areorthogonal(𝒂 ⊥ 𝒃),then𝒂 ∙ 𝒃 = 0
(5) cos 𝜃 = 𝒂 ∙ 𝒃 𝒂 ∙ 𝒃
Scalarproductsoforthonormalbasisvectors𝒆Gand𝒆H
𝒆G ∙ 𝒆G = 𝒆H ∙ 𝒆H = 1
𝒆G ∙ 𝒆H = 𝒆H ∙ 𝒆G = 0
Thesetwofactscanbeneatlysummarisedas
𝒆o ∙ 𝒆p = 𝛿op ,
wheretheKroneckerdeltasymbol𝛿op isdefinedby
𝛿op =1, if𝑖 = 𝑗0, if𝑖 ≠ 𝑗
Algebraicpropertiesofthescalarproduct
𝒂 ∙ 𝒃 = 𝒃 ∙ 𝒂 (commutative)
𝒂 ∙ 𝒃 + 𝒄 = 𝒂 ∙ 𝒃 + 𝒂 ∙ 𝒄 (distributiveovervectoraddition)
Commutativityfollowsimmediatelyfromthedefinition(orfromthecosineformula).DistributivitycanbeverifiedbydrawingprojectionsasinFigures5and6.
Example1.Let𝒂 = 2𝒆G + 5𝒆Hand𝒃 = 3𝒆G − 4𝒆H.Calculate𝒂 ∙ 𝒃.
Solution:Wehave
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𝒂 ∙ 𝒃 = (2𝒆G + 5𝒆H) ∙ (3𝒆G − 4𝒆H) [usedistributivity]
= 2𝒆G ∙ 3𝒆G − 2𝒆G ∙ 4𝒆H + 5𝒆H ∙ 3𝒆G − 5𝒆H ∙ 4𝒆H
= 2 ∙ 3 − 0 + 0 − 5 ∙ 4
= −14.
Noticethatwecouldhavefoundtheanswerbysimplymultiplyingthecoefficientsofthe
respectivebasisvectorsi.e.2 ∙ 3 + 5 ∙ −4 = 6 − 20 = −14.
Coordinateformofthescalarproduct
Let𝒂 = 𝑎G𝒆G + 𝑎H𝒆Hand𝒃 = 𝑏G𝒆G + 𝑏H𝒆H.Then
𝒂 ∙ 𝒃 = 𝑎G𝑏G + 𝑎H𝑏H.
Proof:Justapplydistributivityruleasintheaboveexample.
NB!Thismakesscalarproductahandytoolwhenthevectorsaregiveninanorthonormalbasis.
SCALARPRODUCTINℝMANDHIGHERDIMENSIONS
Thedefinitionofscalarproduct𝒂 ∙ 𝒃(astheproductof 𝒂 and 𝒃𝒂 )extendsnaturallytothree-
dimensionalspacevectors(vectorsinℝM).Alsothecosinerule𝒂 ∙ 𝒃 = 𝒂 ∙ 𝒃 ∙ cos 𝜃isvalid
unchanged.Soisthecoordinateformforthescalarproductofvectorsgiveninanorthonormalbasis
𝒆G, 𝒆H, 𝒆M where𝒆Mispointingtothedirectionof𝑧-axis.Henceif𝒂 = 𝑎G𝒆G + 𝑎H𝒆H + 𝑎M𝒆Mand
𝒃 = 𝑏G𝒆G + 𝑏H𝒆H + 𝑏M𝒆Maretwospacevectorsthen𝒂 ∙ 𝒃 = 𝑎G𝑏G + 𝑎H𝑏H + 𝑎M𝑏M.
Itisworthnoticingthatthecoordinateformgeneralizesnaturallytoanyfinite-dimensionalspace
ℝJwhere𝑛isapositiveinteger.Youjustaddmoreorthonormalbasisvectors𝒆Y, 𝒆v, … , 𝒆Jtocare
ofthenewcoordinateaxes.Thecosineform,however,isnotsoobviousbecausewedonothave
aclearideawhatthenotionofanintermediateangle𝜃canmeaninthesehigher-dimensional
spaces.Indeed,thecosineformcanthenbeusedtodefinetheconceptofanglebyturningthe
cosineformupsidedown:cos 𝜃 = 𝒂 ∙ 𝒃 𝒂 ∙ 𝒃 .Therighthandsidecanbecalculatedby
usingthecoordinateformandrememberingthat 𝒂 = 𝒂 ∙ 𝒂.
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REALNUMBERALGEBRAASASPECIALCASEOFVECTORALGEBRA
AbovewehavedefinedavectoralgebrainℝHwithadditionanddotproductservingasthebasic
operations.WeobservedthatthisalgebraextendsnaturallyintohigherdimensionsℝJ.Butwhat
aboutgoingtothelowerdimension,theusualnumberlineℝ = ℝG.Indeed,wemayconsiderreal
numbersasvectors,thatisvectorspointingfromtheorigintothepointsof𝑥-axiswhichplaysthen
theroleofrealnumberline.Callsuchvectorsrealvectors.Forthesevectorsthereisonlyonebasis
vector,namely𝒆G,andeveryrealvectorisoftheform𝑎𝒆G,where𝑎isarealnumber(justthe
numberthevector𝑎𝒆Gispointingto).Iteasytoseethatadditionoftworealvectors𝑎𝒆Gand𝑏𝒆G,
gives(𝑎 + 𝑏)𝒆Gwhichpointstotherealnumber𝑎 + 𝑏,infullcoherenceoftheadditionorreal
numbers.Likewisethescalarproductof𝑎𝒆Gand𝑏𝒆Ggives(usingthecoordinateformula)𝑎𝑏,the
usualproductofrealnumberstowhichthesetwovectorsarepointingto.
WHYNOTSTOPHERE?
Well,youmightthink,nowwehavethescalarproduct(dot),anicevectormultiplication
operation,easytocalculateinvariouswaysandworkingfine(distributively)withaddition.Itis
alsophysicallymeaningful(e.g.incalculatingtheworkdonebyaforce).Whydon’twestophere
andbuildourvectoralgebra(inanydimension)ontheoperationsofaddition𝒂 + 𝒃andscalar
product𝒂 ∙ 𝒃?
True,wenowhaveaveryusefulalgebraicmachineryatourdisposalandalotcanbedonewithit.
However,mathematically,wecannotbefullysatisfiedwiththeconceptofscalarproduct.We
wouldliketohaveawayofmultiplyingvectorswheretheanswersarealsovectors.Thisisnotthe
caseforthescalarproductwheretheanswersarerealnumbers.Therearealsophysicalsituations
(particularlyinstudyingelectromagnetism)whereitwouldbeuseful,evennecessary,toget
vectoranswers.Thereforewecontinueoursearchforothervariantsofmultiplyingvectors.And
thisturnsouttobeafascinatingexcursion!
16
5.COMPLEXPRODUCT𝒂 ∗ 𝒃inℝ𝟐
Wenowintroduceoursecondversionofvectormultiplicationwhichwecallcomplexproductor
rotationalproduct(althoughthisisnostandardterminology).ThisproductisdefinedonlyinℝH,
thatisforplanevectors𝒂 = 𝑎G𝒆G + 𝑎H𝒆Hand𝒃 = 𝑏G𝒆G + 𝑏H𝒆H.Likethescalarproduct,alsothe
complexproduct(denotedhereby𝒂 ∗ 𝒃,againnostandardnotation)canbedefinedintwo
differentways:geometricallyintermsoflengthsanddirectionsof𝒂and𝒃oralgebraicallyin
termsoftheirexpressionsinthebasis 𝒆G, 𝒆H .
GEOMETRICDEFINITIONOF𝒂 ∗ 𝒃
ThedefinitionisexplainedinFigure7below.
Figure7.Thecomplexproductoftheplanevectors𝒂and𝒃.Thedirectionsofthesevectorsaregivenbytheangles𝛼and𝛽theymakewithafixedaxisofreference(whichmaybeassumedtobethe𝑥-axisorrealaxis).Thelengthoftheproductvector𝒂 ∗ 𝒃isequaltotheproductofthelengthsof𝒂and𝒃.Thatis 𝒂 ∗ 𝒃 = 𝒂 ∙ 𝒃 .Thedirectionangleoftheproductvector𝒂 ∗ 𝒃isthesumofthedirectionangles𝛼and𝛽.Theproductvector𝒂 ∗ 𝒃sitsinthesameoriginasthefactorvectors𝒂and𝒃butisshownhereseparatelyforthesakeofvisualclarity.(Noticethatifthevectors𝒂and𝒃arebothparalleltotheaxisofreference,thatisalongtheaxis,thenthedirectionangles𝛼and𝛽areeither0°or180°andthecomplexproductvector𝒂 ∗ 𝒃isalongthesameaxis.Thissituationcorrespondstotheusualmultiplicationofrealnumbers.)
17
Algebraicpropertiesofthecomplexproduct
𝒂 ∗ 𝒃 = 𝒃 ∗ 𝒂 (commutative)
𝒂 ∗ 𝒃 + 𝒄 = 𝒂 ∗ 𝒃 + 𝒂 ∗ 𝒄 (distributiveovervectoraddition)
𝑡𝒂 ∗ 𝒃 = 𝒂 ∗ 𝑡𝒃 = 𝑡(𝒂 ∗ 𝒃) (freemobilityofthescalarfactor𝑡)
Commutativityandscalarfactorruleareself-evidentfromtheabovedefinition.Distributivitycan
alsobedemonstratedbyacarefuldrawingexercise.
Asinthecaseofthescalarproductitisagainusefultoexpressthecomplexproduct𝒂 ∗ 𝒃inthe
coordinateform,thatisintermsoforthonormal12basisvectors𝒆Gand𝒆H.Soweassumeagain
thatourvectors”live”intheusualCartesiancoordinatesystemwithorthogonal𝑥-and𝑦-axes
whichareparalleltothebasisvectors.Wealsoassumethatthe𝑥-axisistheaxisofreference(also
calledtherealaxis)againstwhichalldirectionanglesaremeasured.Hencethe𝑥-directedbasis
vector𝒆Grepresentstherealnumber1andgenerallythevector𝑎𝒆Grepresentstherealnumber𝑎.
Complexproductsoftheorthonormalbasisvectors
𝒆G ∗ 𝒆G = 𝒆G
𝒆H ∗ 𝒆H = −𝒆G
𝒆G ∗ 𝒆H = 𝒆H ∗ 𝒆G = 𝒆H
Theseresultsfollowimmediatelyfromtheabovedefinition(Figure7)becausethelengthsofthe
basisvectorsare 𝒆G = 𝒆H = 1andtheirdirectionanglesare0°and90°.
Coordinateformofthecomplexproduct
Considertheplanevectors𝒂 = 𝑎G𝒆G + 𝑎H𝒆Hand𝒃 = 𝑏G𝒆G + 𝑏H𝒆H.Then 𝒂 ∗ 𝒃 = 𝑎G𝑏G − 𝑎H𝑏H 𝒆G + 𝑎G𝑏H + 𝑎H𝑏G 𝒆H
Thisresultfollowseasilyfromthepreviousboxusingdistributivityrule.
Example2.Let𝒂 = 3𝒆G − 2𝒆Hand𝒃 = 7𝒆G + 9𝒆H.Calculatethecomplexproduct𝒂 ∗ 𝒃.
12Rememberthatorthonormalvectorsareperpendicularunitvectors.
18
Solution:Bythecoordinateformulawehave
𝒂 ∗ 𝒃 = 3 ∙ 7 − (−2) ∙ 9 𝒆G + 3 ∙ 9 + (−2) ∙ 7 𝒆H
= 39𝒆G + 15𝒆H.
�
AbovewehaveusedtheCartesian13coordinates𝑥and𝑦toidentifyeachpointP= (𝑥, 𝑦)inthe
planeℝH.Thecorrespondingpositionvector(fromtheoriginOtoP)isthenOP=𝑥𝒆G + 𝑦𝒆H.Itis
oftenusefultointroducepolarcoordinateswhichidentifyeachpointPoftheplanebyitsradius
anddirection.Theradius𝑟isequaltothedistanceofPfromtheoriginO,i.e.thelengthofthe
vectorOP.Thedirection𝜃isthedirectionangleofthisvector,thatistheanglebetweenOPand
the𝑥-axis(seeFigure8).
Figure8.ThepointPhasCartesiancoordinates(𝑥, 𝑦)andpolarcoordinates(𝑟, 𝜃).Fromtherighttrianglesweseethat𝑥 = 𝑟 cos 𝜃and𝑦 = 𝑟 sin 𝜃aswellas𝑟H = 𝑥H + 𝑦Handtan 𝜃 = 𝑦/𝑥.Thepositionvector(inred)OP=𝑥𝒆G + 𝑦𝒆H = 𝑟 cos 𝜃 𝒆G + 𝑟 sin 𝜃 𝒆H = 𝑟(cos 𝜃 𝒆G + sin 𝜃 𝒆H).
COMPLEXPRODUCTINPOLARCOORDINATES
Polarcoordinatesareidealforcalculatingthecomplexproductsofvectors.Assumethatthe
vector𝒂haspolarcoordinates(𝑟G, 𝜃G)andthevector𝒃haspolarcoordinates(𝑟H, 𝜃H).Itfollows
fromthedefinition(Figure7)thattheradius(length)ofthecomplexproduct𝒂 ∗ 𝒃isavector
whoselength(radius)isequalto𝑟 = 𝒂 ∙ 𝒃 = 𝑟G𝑟Handitsdirectionangle𝜃 = 𝜃G + 𝜃H.So𝒂 ∗ 𝒃
haspolarcoordinates 𝑟, 𝜃 = (𝑟G𝑟H, 𝜃G + 𝜃H).
13SonamedaftertheFrenchmathematicianRenéDescartes(1596-1650),theinventorofcoordinatesystems.HisLatinnamewasCartesius.
19
Thisfactcanbeusedtoprovetrigonometrictheorems.Take𝒂and𝒃tobeunitvectors,thatis
𝑟G = 𝑟H = 1withdirectionangles𝜃Gand𝜃H.Sowehave(asexplainedinFigure8)that𝒂 =
cos 𝜃G 𝒆G + sin 𝜃G 𝒆Hand𝒃 = cos 𝜃H 𝒆G + sin 𝜃H 𝒆H.Then𝒂 ∗ 𝒃isalsoaunitvector(𝑟 = 𝑟G𝑟H =
1)anditsdirectionangleis𝜃 = 𝜃G + 𝜃H.Thenwecanalsoexpresstheproductvector𝒂 ∗ 𝒃in
termsofthebasisvectorsintheform
𝒂 ∗ 𝒃 = cos 𝜃 𝒆G + sin 𝜃 𝒆H = cos(𝜃G + 𝜃H) 𝒆G + sin(𝜃G + 𝜃H) 𝒆H.
ButwecanalsocalculatethesameproductbythedistributiveruleasaboveinExample2:
𝒂 ∗ 𝒃 = cos 𝜃G 𝒆G + sin 𝜃G 𝒆H ∗ (cos 𝜃H 𝒆G + sin 𝜃H 𝒆H)
= (cos 𝜃G cos 𝜃H − sin 𝜃G sin 𝜃H) 𝒆G + (cos 𝜃G sin 𝜃H + sin 𝜃G cos 𝜃H) 𝒆H.
Thetworesultsmustbeidentical.Thereforewehave
cos(𝜃G + 𝜃H) = cos 𝜃G cos 𝜃H − sin 𝜃G sin 𝜃H
sin(𝜃G + 𝜃H) = cos 𝜃G sin 𝜃H +sin 𝜃G cos 𝜃H
whicharethewell-knownadditionformulaetobefoundinbooks(e.g.MAOL).
COMPLEXNUMBERS
Historicallytheideaofcomplexproductwasinventedlongtimebeforetheconceptofvectorwas
around14.Abovewehavedefinedthecomplexproductasanoperationforvectors.In
mathematics,however,thecomplexproductisusuallyseenasanoperationinthesetℂof
complexnumberswhichisanextensionofthefamiliarsetℝofrealnumbers.Whilethesetℝis
geometricallyrepresentedbypointsofthenumberlinethesetℂisrepresentedbypointsofthe
numberplaneofwhichtheℝ-lineisasmallpartlike𝑥-axisisbutasmallpartofthewhole𝑥𝑦-
14ItmayhavebeenthoughtalreadybytheancientGreeksbutItalianmathematiciansRafaelBombelli(1526-1572)andGirolamoCardano(1501-1576)certainlyuseditinthestudyofequations.Forthemitwasanoperationforanextendedsetofnumbers(socalledcomplexnumbers,lateridentifiedwithpointsofthexy-plane).TheconceptofvectorwasintroducedonlythreecenturieslaterbytheIrishmathematicianWilliamRowanHamilton(1805-1865)andotherseventhoughCarlFriedrichGauss(1777-1855)depictedcomplexnumbersas”arrows”fromtheorigintothepointsofplane.
20
plane.Thisrepresentationcanbeextendedtofurthervectorssincepointsand(position)vectors
areessentiallyidentical.
Intheoriginalcomplexnumberterminologybasisvectorsarenotusedbut𝒆Gissimplywrittenas
therealnumber1.Theotherbasisvector𝒆Hpointstoanon-realpoint(0,1)onthe𝑦-axisandthis
pointisthoughttorepresentanewkindofnon-realnumberunitdenotedbythesymbol𝑖which–
forhistoricalreasons–iscalledtheimaginaryunit.
Inthisapproachtheplanevector𝒛 = 𝑥𝒆G + 𝑦𝒆Hisreplacedbythe”number”𝑧 = 𝑥 + 𝑦𝑖.These
numbersarecalledcomplexnumbers–complexbecausetheyarecombinations(orcomplexes)of
arealnumber𝑥andanimaginarynumber𝑦𝑖.Therealnumbers𝑥and𝑦aresimplycoordinatesof
thepointwherethenumber𝑧 = 𝑥 + 𝑦𝑖islocated.Theabovementionedproductrulesforthe
basisvectorscannowbetranslatedintorulesforcomplexnumberalgebrawhichturnsoutto
workexactlyasthealgebraofrealnumbers.Theonlynewruleisthat𝑖H = −1whichisjusta
complexnumberlanguageversionofourvectorrule:𝒆H ∗ 𝒆H = −𝒆G.(Inwritingtheequation
𝑖H = −1wehaveomitted*asthesymbolofcomplexproduct.Insteadof𝑖 ∗ 𝑖wewritesimply
𝑖𝑖 = 𝑖Hfollowingthefamiliarnotationfortheproductoftwonumbers.)
Asincaseofvectorswecanwriteanycomplexnumber𝑧inCartesianorpolarform
𝑧 = 𝑥 + 𝑦𝑖 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃).
Thediscoveryofcomplexnumbershasopeneddoorstoanewandextremelyrichterritoryof
mathematicswhichhasturnedintoanindispensabletoolformodernphysics.
EULEREQUATION
AsanexampleofthetreasuresofcomplexnumberswederivethefamousequationofLeonhard
Euler15whichshowsthatthetrigonometricfunctionscos 𝑥andsin 𝑥andtheexponentialfunction
𝑒|arecloserelatives–afactwhichremainsundetectedifweallowthevariable𝑥tohaveonly
realnumbervalues.Letting𝑥freetorangethecomplexplanemakesthiskinshipapparent.
15LeonhardEuler(1707-1783),aSwissmathematician,probablythemostproductiveofalltime.ThepublicationofhisOperaomnia(Collectedworks)beganin1911andtothisdate(2015)already76thickvolumeshavebeenprinted.
21
Itisawellknownfactthatfortherealvaluesof𝑥thesefunctionscanbeexpressedasinfinite
polynomials,calledseriesexpansions,whichalsoprovidethealgorithmsforthecalculatorsto
respondwhenyoupressthesefunctionbuttons:
𝑒| =𝑥J
𝑛!
~
�
= 1 + 𝑥 +𝑥H
2 +𝑥M
6 +𝑥Y
24 +𝑥v
120 +𝑥�
720 +𝑥�
5040 +⋯
cos 𝑥 =𝑥HJ
(2𝑛)! ∙ −1J
~
�
= 1 −𝑥H
2 +𝑥Y
24 −𝑥�
720 +−⋯
sin 𝑥 =𝑥HJ�G
(2𝑛 + 1)! ∙ −1J
~
�
= 𝑥 −𝑥M
6 +𝑥v
120 −𝑥�
5040 + −⋯
Inthetwolatterexpansionsitisassumedthat𝑥isanangleinradians(notindegrees).
Wecannowallowcomplexnumbervaluesfor𝑥becausecalculatingpolynomialsinvolvesonly
additionandmultiplicationwhich(forcomplexnumbers)followexactlythesamerulesasforreal
numbers.Soletusput𝑥 = 𝜃𝑖where𝜃isarealnumberand𝑖istheimaginaryunit(anothername
fortheunitbasisvector𝒆H).Thencalculatethepowersof𝑥appearingintheaboveexpansions
rememberingthat𝑖H = −1.Wehave
𝑥� = 𝜃𝑖 � = 1
𝑥G = 𝜃𝑖 G = 𝜃𝑖
𝑥H = 𝜃𝑖 H = −𝜃H
𝑥M = 𝜃𝑖 M = −𝜃M𝑖
𝑥Y = 𝜃𝑖 Y = 𝜃Y
𝑥v = (𝜃𝑖)v = 𝜃v𝑖
etc.
Substitutingthesetotheexpansionof𝑒|andrearrangingthetermsintotwogroups(evenpowers
andoddpowers)wenoticethatthetheevenpowersconstituteexactlytheexpansionofcos 𝜃
andtheoddpowersrespectivelyconstitutetheexpansionofsin 𝜃 ∙ 𝑖 = 𝑖 sin 𝜃.Sowehave
𝑒o� = cos 𝜃 + 𝑖 sin 𝜃.
ThisistheEulerequationwhichshowsexponentialandtrigonometricfunctionsascloserelatives.
Inparticular,for𝜃 = 𝜋,weget
22
𝑒o� = −1or𝑒o� + 1 = 0
whichissometimescalledthemostbeautifulequationofmathematicsbecauseitconnectsthe
fiveimportantmathematicalconstants:0, 1, 𝜋, 𝑒and𝑖.
NotethattheEulerequationallowsustowritethepolarcoordinateformofacomplexnumber
veryconciselyasfollows
𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃 = 𝑟𝑒o�.
Thisissometimescalledtheexponentialformof𝑧butessentiallyitisthepolarformwith𝑟and𝜃
showninsteadofCartesiancoordinates𝑥and𝑦.
THENORMOFACOMPLEXNUMBER
Consideracomplexnumber𝑧 = 𝑥 + 𝑦𝑖.Thenon-negativerealnumber 𝑥H + 𝑦Hiscalledthe
norm(orabsolutevalue)of𝑧anddenotedby 𝑧 .Itisequaltothedistanceofthepoint𝑧 = (𝑥, 𝑦)
fromtheorigin,orinvectorterminology,thelengthofthepositionvectorofthatpoint.
ByEulerequationwehave 𝑒o� = cos 𝜃 H + sin 𝜃 H = 1whichimpliesthatthecomplex
number𝑒o�always(thatis,forallvaluesof𝜃)liesontheorigin-centredunitcircle.Thevalueof𝜃
indicatesthedirectionangleofthepoint(orthevector)𝑒o�inradians(seeFigure9).
Figure9.Thecomplexnumber𝑒o�shownasavector.As𝜃increasesfrom0to2𝜋(inradians)thepoint𝑒o�movesaroundtheorigin-centredunitcircle.Thenormof𝑒o�isalwaysequaltoone.
23
ROTATIONINTHECOMPLEXPLANE
Let𝑧beacomplexnumber,thatisapoint(vector)inthecomplexplane.Ifyouwanttorotatethis
pointbyanangle𝜃abouttheoriginyoucandoitsimplybycalculatingthecomplexproduct𝑒o�𝑧.
Bydefinitionthecomplexmultiplicationmeansmultiplyingthelengths(norms)ofthevectors𝑒o�
and𝑧.Theformeris1andhencethelength(norm)oftheproductisequaltothenormof𝑧.Hence
𝑒o�𝑧liesonthesameorigin-centredcircleas𝑧.Furthermore,bythedefinitionagain,thedirection
angleofthecomplexproduct𝑒o�𝑧isobtainedbyaddingthedirectionanglesof𝑒o�(whichis𝜃)
and𝑧.Hencethepoint(orvector)correspondingtheproduct𝑒o�𝑧isobtainedbyrotatingthe
point(orvector)correspondingto𝑧byanangle𝜃abouttheorigin(seeFigure10).
Figure10.Multiplying𝑧bythecomplexfactor𝑒o�rotatesthepoint(vector)𝑧bytheangle𝜃.Theoriginalpoint𝑧andtherotatedpoint𝑒o�𝑧areonthesameorigin-centredcircle.Thenormsofthesetwocomplexnumbersareequal: 𝑧 = 𝑒o�𝑧 .
WHYNOTSTOPHERE?
Thecomplexproduct𝒂 ∗ 𝒃seemsanidealcandidateforthemultiplicationofvectors.Together
withvectoraddition𝒂 + 𝒃itsatisfiesallrules(associativity,commutativity,distributivityetc)that
arevalidforrealnumbersandmaketheiralgebrasuchaneffectivetoolforavastrangeof
applicationsinsciencesandevenineverydaylife.Whythenshouldweseekanyotherwaysof
definingthemultiplicationofvectors?
True,complexalgebrafortheplanevectors(orcomplex”numbers”astheyarecustomarilycalled
inthiscase),isalsoaverybeautifulandextremelyeffectivetoolinmanyareas,particularlyin
physics.Butthereisacruciallimitation:itworksonlyintwo-dimensionalcaseforplanevectorsin
24
ℝH.Thisissobecausethecomplexproduct𝒂 ∗ 𝒃isessentiallyarotationintheplane.However,
wewouldliketohavevectoralgebrawhichwouldgeneralizetohigherdimensionsandparticularly
toℝMwhichisthemathematicalcounterpartofourthree-dimensionalspace.That’swhywe
continueoursearchfordifferentwaysofdefiningtheproductoftwovectors.(Fortheadditionof
vectorswealreadyfoundanaturalandwell-functioningdefinitionwhichgeneralizestoall
dimensionsinℝJ.)
25
6.VECTORPRODUCTORCROSSPRODUCT𝒂×𝒃
PHYSICALBACKGROUND
Theideaofthecrossproductoftwovectorswasfirstconveivedbyphysicistswhoneededa
mathematicalconcepttodescribenotionsliketorque(twistormoment16)causedbyaforcewhich
makesanobjectrotateaboutafixedaxis(seeFigure11).
Figure11.AbarOAcanrotatearoundthefixedaxispointO.TherotationiscausedbyaforceonthepointAofthebar.Thisforcevector𝑭makesanangle𝜃withthedirectionOA.Thetorque(twistingeffect)isduetotheforcecomponent𝑭� = 𝑭 sin 𝜃(inred)whichisperpendicularto𝒓,thepositionvectorofA.Fromphysicalexperienceweknowthatthemagnitudeofthetorqueisproportionaltothemagnitudeof𝑭�andandthemagnitudeof𝒓.Thereforeitisnaturaltodefinethetorquevector𝑴asavectorwhosemagnitudeis 𝑴 = 𝑭� ∙ 𝒓 = 𝑭 ∙ 𝒓 ∙ sin 𝜃.Thedirectionoftorquevectorisdefinedtobeperpendiculartotheplaneofrotation.(Itcannotbeintheplanebecausethedirectionsof𝑭and𝒓(whichareintheplane)areconstantlychangingduringtherotation.Thevector𝑴iscalledthecrossproductof𝑭and𝒓,insymbols𝑴 = 𝑭×𝒓.
ItisclearfromtheFigure11thattheoperationofcrossproductrequiresathree-dimensional
spaceℝMasitsbackgroundenvironment.Theproductvector𝑭×𝒓isnotinthesameplanewithits
factorvectors𝑭and𝒓.
Besidesthetorque𝑴therearemanyotherphysicalconceptswhichareaptlydescribedbycross
productstwovectors.Examplesincludeangularmomentum17𝑳 = 𝒓×𝒑inmechanicsandLorentz-
force18𝑭 = 𝒗×𝑩inelectromagnetism.
16Nottobeconfusedwithmomentum(masstimevelocity)ofamovingparticle.17Angularmomentummeasuresthe”amount”ofrotationalmotioninthesamewayasmomentummeasuresthe”amount”oftranslationalmotion.18Forceonachargedparticlemovingatvelocity𝒗inthemagneticfieldofstrength𝑩.
26
DEFINITIONOFTHECROSSPRODUCTANDITSBASICPROPERTIES
Let𝒂and𝒃andbetwovectorsinℝMwhichmakeanangle𝜃witheachother.Thecrossproduct
𝒂×𝒃isdefinedasavectorwhosenorm(absolutevalueormagnitude)is 𝒂×𝒃 = 𝒂 ∙ 𝒃 ∙ sin 𝜃,
which,bytheway,istheareaoftheparallelogramwithsides𝒂and𝒃.Thedirectionof𝒂×𝒃is
definedtobeperpendiculartotheplaneof𝒂and𝒃suchthatthethreevectors𝒂, 𝒃and𝒂×𝒃
formaright-handsystem19.
Thecrossproductisalsocalledvectorproducttoemphasizethefactthattheresult𝒂×𝒃isa
vector(andnotascalarasincaseofthedotproduct𝒂 ∙ 𝒃).
ALGEBRAICPROPERTIESOFCROSSPRODUCT
Itisfairlyeasytoseethatthecrossproductsatisfiesthefollowingrules:
𝒂×𝒃 = −𝒃×𝒂 (anticommutativity)
𝒂× 𝒃 + 𝒄 = 𝒂×𝒃 + 𝒂×𝒄 (distributivity)
𝑡𝒂 ×𝒃 = 𝒂× 𝑡𝒂 = 𝑡(𝒂×𝒃) (freemobilityofascalarfactor)
COORDINATEFORMOFTHECROSSPRODUCT
Let 𝒆G, 𝒆H, 𝒆M beanorthonormal20basisforℝM.Bytheabovedefinitionwecandeterminethe
crossproductsofthesebasisvectors.
𝒆G×𝒆G = 𝒆H×𝒆H = 𝒆M×𝒆M = 𝟎
𝒆G×𝒆H = 𝒆M,𝒆H×𝒆G = −𝒆M
𝒆H×𝒆M = 𝒆G,𝒆M×𝒆H = −𝒆G
𝒆M×𝒆G = 𝒆H,𝒆G×𝒆M = −𝒆H
19Threevectors𝒂, 𝒃and𝒄formaright-handsystemifyoucanalignthethreerighthandfingers(forefinger𝒂,middlefinger𝒃andthumb𝒄)pointingintheirdirections.Theleft-handsystemisdefinedanalogously.ItisatheoremofspacegeometrythateverytripleofvectorsinℝMiseitherright-handsystemoraleft-handsystemandneverboth.Alsoifyouturnoneofthetriple’svectorstooppositedirection(e.g.change𝒃to– 𝒃)thenthehandednessofthetriplechangesfromrighttoleftorviceversa.20Rememberthatorthonormalmeansperpendicular(orthogonal)andofunitnorm.
27
Assumenowthatarbitraryvectors𝒂and𝒃aregiveninthecoordinateformintermsof
orthonormalbasisvectors 𝒆G, 𝒆H, 𝒆M asfollows 𝒂 = 𝑎G𝒆G + 𝑎H𝒆H + 𝑎M𝒆M,
𝒃 = 𝑏G𝒆G + 𝑏H𝒆H + 𝑏M𝒆M.Wecannowusetheaboveobservationstocomputethecrossproductof𝒂and𝒃through
termwisemultiplication.Aroutinecalculationgives:
𝒂×𝒃 = 𝑎H𝑏M − 𝑎M𝑏H 𝒆G + 𝑎M𝑏G − 𝑎G𝑏M 𝒆H + 𝑎G𝑏H − 𝑎H𝑏G 𝒆M
Youcaneasilycheckthisresultbycalculatingthedotproducts𝒂 ∙ (𝒂×𝒃)and𝒃 ∙ (𝒂×𝒃).Bothwill
turnouttobezerowhichprovesthatourcalculated𝒂×𝒃isperpendicularagainst𝒂and𝒃asit
shouldbe!Checkingthecorrectnessofthenorm(length)isabitmoretediouscalculation(you
havetoevaluatesineoftheanglebetweenthevectors)butstillfeasible.
Exercise:Calculatethecrossproductofthevectors𝒂 = 𝒆G + 2𝒆H − 2𝒆Mand𝒃 = 2𝒆G − 2𝒆H − 𝒆M.Verifyyourresultbycalculatingthelengthsofrelevantvectorsandstretchingthefingersofyourrighthand.WHYNOTSTOPHERE?
Onceagainwehavedrawnfromphysicalintuitionanddefinedanewandinteresting(andvery
useful)wayofmultiplyingvectors:thecrossproduct.Fromtwovectorsitproducesavectorand
thisisanimprovementcomparedtodotproductwhichproducesascalar.Thecrossproductalso
worksfineinthethree-dimensionalspaceℝM-animprovementcomparedtocomplexproduct
whichworksonlyinℝH.
However,thecrossproducthasalsoitsdrawbacks.Aminoroneisthatcommutativityfails
althoughanticommutativitycompensatesitinanaturalway.Changingtheorderoffactorsdoes
notcauseanyhazardouschangesintheproduct.Amoreseriousproblemaboutthecrossproduct
isthatitactuallyworksonlyinℝM.Youcancross-multiplyvectorsinℝHbuttheresultsstickoutto
higherdimensionℝM.Butcross-multiplyingvectorsindimensionshigherthanthree(thatis,
ℝY,ℝv, ℝ�, ….)causeevenharderproblemsbecauseinℝY,forexample,thedirectionof𝒆G×𝒆H
cannotbeuniquelydefined.InℝYthereareseveralnon-parallelvectors(𝒆Mand𝒆Yamongothers)
whichareperpendiculartoboth𝒆Gand𝒆H.Sothesearchgoeson!
28
29
7.QUATERNIONPRODUCT𝒂 ⋄ 𝒃INℝ𝟒
WilliamRowanHamilton(1805-1865)wasveryimpressedabouttheeleganceandefficiencyof
complexnumbers(thatis,vectorsinℝ𝟐equippedwiththeoperationsofadditionandcomplex
product).Hedreamedofthepossibilityofgeneralizingthecomplexalgebraofℝ𝟐tohigher
dimensions,particularlytoℝ𝟑andworkedalongtimetofindanappropriatedefinitionof
”complexproduct”forthree-dimensionalvectors.Hefailedagainandagainuntilitsuddenly
occurredtohim21tomoveontoℝ𝟒wherethepiecesofhismazemiraculouslycametogether.He
hadinventedthequaternionproduct.
Inthecomplexplaneℂ(whichisessentiallythetwo-dimensionalvectorspaceℝHequippedwith
additionandcomplexmultiplication)wehavetwoaxes:therealaxisrepresentingthereal
numbers𝑥andtheimaginaryaxisrepresentingimaginarynumbers𝑦𝑖,wheretheimaginaryunit𝑖
isdefinedasanumbersatisfyingthecondition𝑖H = −1.Allpoints(𝑥, 𝑦)oftheplane,taken
together,representcomplexnumbers𝑥 + 𝑦𝑖.HamiltontriedtoextendthistoℝMbyintroducinga
thirdaxis(𝑧)asasecondimaginaryaxisrepresentingnumbers𝑧𝑗(againwith𝑗H = −1but𝑗 ≠ 𝑖).
Inthissettingthepoint(𝑥, 𝑦, 𝑧)ofℝMwouldberepresentedbyanumber𝑥 + 𝑦𝑖 + 𝑧𝑗.Itturned
out,however,thatitwasimpossibletodefinethemultiplicationofsuch”hypercomplex”numbers
inareasonableway.
InhisBroombridgeheureka-momentHamiltonrealizedthatallproblemswouldsortoutneatlyby
addingthefourthaxisonwhichthethirdimaginaryunit𝑘wouldbeliving.Ineffectthismeans
steppingfromℝMtoℝY.Inacloserinspectionitturnedoutconvenienttonamethethree
imaginaryaxesastheusualspacecoordinateaxes𝑥, 𝑦and𝑧andkeeptherealaxis𝑡separate22.
21Thelegendhasitthattherevelationcamein1843whenHamiltonwaswalkingovertheBroombridgeinDublin.Hewassoexcitedaboutitthathecarvedsomecrucialequationsinthebridgestones.Thecarvingsaresaidtobestillvisiblethere.LaterithasbeenfoundthataFrenchmathematician(andbanker)OlindeRodrigues(1795–1851)hadmadeessentiallythesameinventionin1840buthisworkremainedunknownforseveraldecades.22Symbol𝑡fortherealaxismaybeseenasahintthatthisaxismightphysicallyrepresenttimewhilethreeotheraxesrepresentspatialposition.Insomeapplications(relativitytheory)thisindeedisacase.Generallythe𝑡-axisissimplyascalaraxiswhichrepresentsrealnumbers.
30
Imitatingthecomplexnumbernotationwenowhave”numbers”oftheform𝑡 + 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘.
Geometricallythesenumbers(whichHamiltonnamedquaternions23)arerepresentedbypoints
(𝑡, 𝑥, 𝑦, 𝑧)inℝYorequivalentlybyvectors𝑡𝒆� + 𝑥𝒆G + 𝑦𝒆H + 𝑧𝒆Mwhere𝒆�, 𝒆G, 𝒆Hand𝒆Mare
orthonormalbasisvectorsforℝY.
DEFINITIONOFQUATERNIONSANDTHEQUATERNIONPRODUCT
IntheHamiltonianapproachquaternionsare4-dimensionalmathematicalobjects–sortof
extendedcomplexnumbers–oftheform𝑞 = 𝑡 + 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘,wherethefirstterm𝑡iscalled
thescalar(orreal)partof𝑞andthesum𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘istheimaginarypart(alsocalledthevector
part)of𝑞.Quaternionscanbeaddedandmultipliedbyscalarsasusual.Anytwoquaternionscan
alsobemultipliedintheusualway(obeyingtherulesofassociativity,distributivityandfree
mobilityofscalarfactors)toproducearesultcalledquaternionproduct.Inthemultiplication
processyouhavetoobservethefollowingspecialruleregardingtheimaginaryunits𝑖, 𝑗and𝑘:
𝑖H = 𝑗H = 𝑘H = 𝑖𝑗𝑘 = −1.
Intheseidentitieswehavefollowed(aswedidearlierforcomplexnumbers)theconventionof
writingthequaternionproductswithoutanymultiplicationsymbollike.Soinsteadof𝑖 ⋄ 𝑖wewrite
simply𝑖𝑖 = 𝑖Hetc.
Thesetofallquaternions𝑞 = 𝑡 + 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘issometimesdenotedbythesymbolℍ(inhonour
ofHamilton).
PRODUCTSOFIMAGINARYUNITS 1a 𝑖𝑗 = 𝑘, 1b 𝑗𝑖 = −𝑘
2a 𝑗𝑘 = 𝑖, 2b 𝑘𝑗 = −𝑖
3a 𝑘𝑖 = 𝑗, 3b 𝑖𝑘 = −𝑗
Proof:Bydefinitionwehavetheidentity𝑖𝑗𝑘 = −1.Multiplyingbothsidesfromtheright24by𝑘we
get𝑖𝑗𝑘𝑘 = −𝑘.Butbydefinitionagain𝑘𝑘 = 𝑘H = −1andhence𝑖𝑗(−1) = −𝑘.Finallymultiplying
23Bycallingthemquaternions(fromLatinquattuor=four)Hamiltonprobablyreferredtothefactthattheyhavefourcomponents.24Inthedefinitionofthequaternionproductwedidnotassumecommutativity.Thereforewehavetobecarefulwiththeorderofmultiplication.
31
bothsidesby(−1)andusingthefreemobilityruleforscalarfactorsweget𝑖𝑗 = 𝑘.Thisshows
thattheidentity(1a)followsfromthedefinitionofthequaternionproduct.Nowmultiplyingthis
identityfromtheleftby𝑖gives𝑖𝑖𝑗 = 𝑖𝑘or−𝑗 = 𝑖𝑘whichistheidentity(3b).Againmultiplying
thisidentityfromtherightby𝑘gives−𝑗𝑘 = 𝑖𝑘𝑘or−𝑗𝑘 = −𝑖or𝑗𝑘 = 𝑖whichis(2a).Then
multiplying(1a)fromtherightby𝑗gives𝑖𝑗𝑗 = 𝑘𝑗or−𝑖 = 𝑘𝑗whichis(2b).Multiplying(2b)from
therightby𝑗gives−𝑖𝑗 = 𝑘𝑗𝑗or−𝑖𝑗 = −𝑘or𝑖𝑗 = 𝑘whichis(1b).Finallymultiplying(2b)from
therightby𝑘gives−𝑘𝑖 = 𝑘𝑘𝑗or−𝑘𝑖 = −𝑗or𝑘𝑖 = 𝑗whichis(3a).Sowehaveshownthatallsix
identitiesfollowfromthedefinitionofquaternionproduct.QED.
QUATERNIONPRODUCTISNEITHERCOMMUTATIVENORANTICOMMUTATIVE
Fromtheproductsofimaginaryunitsweseethatthequaternionproductiscertainlynot
commutative.Theresults,infact,suggestthatitmightbeanticommutativelikethecrossproduct.
Buteventhatisonlytrueforsomespecialquaternions(liketheimaginaryunits)butnotgenerally.
Takeforexamplethequaternions𝑞G = 1 + 𝑖and𝑞H = 1 + 𝑗.Thenwehave
𝑞G𝑞H = 1 + 𝑖 1 + 𝑗 = 1 + 𝑖 + 𝑗 + 𝑖𝑗 = 1 + 𝑖 + 𝑗 + 𝑘,
but
𝑞H𝑞G = 1 + 𝑗 1 + 𝑖 = 1 + 𝑖 + 𝑗 + 𝑗𝑖 = 1 + 𝑖 + 𝑗 − 𝑘.
Weseethat𝑞G𝑞H ≠ 𝑞H𝑞Gand𝑞G𝑞H ≠ −𝑞H𝑞G.Hencebothcommutativityandanticommutativity
failgenerallyforthequaternionproduct.Thereforewehavetobeverycarefulwiththeorderof
factorswhendoingquaternionproducts.Inspiteofthisobviousdrawbackquaternionshave
turnedoutveryusefultoolsinphysicsandcomputergraphics.
NOTE:Noticethatthatthesubsetofquaternions𝑡 + 𝑥𝑖inwhich𝑦 = 𝑧 = 0(thatis, 𝑗and𝑘do
notappear)isanexactcopyofthesetℂofcomplexnumbers.Ifweoperateonlywithquaternions
inthissubsetthen,ofcourse,multiplicationiscommutative.Thesameistrueforsymmetric
subsetswhere𝑥 = 𝑧 = 0(𝑖and𝑘donotappear)or𝑥 = 𝑦 = 0(𝑖and𝑗donotappear)whichboth
areequivalentwithℂ.
32
QUATERNIONPRODUCTVS.DOTANDCROSSPRODUCTS
Letusconsiderpurevectorquaternions25
𝒂 = 𝑎G𝑖 + 𝑎H𝑗 + 𝑎M𝑘,
𝒃 = 𝑏G𝑖 + 𝑏H𝑗 + 𝑏M𝑘,
andcalculatetheirquaternionproduct𝒂 ⋄ 𝒃observingtheknownproductsof𝑖, 𝑗and𝑘.Wehave
𝒂 ⋄ 𝒃 = (𝑎G𝑖 + 𝑎H𝑗 + 𝑎M𝑘)(𝑏G𝑖 + 𝑏H𝑗 + 𝑏M𝑘)
=...multiplytermwiseandregroup(exercise)...
= −(𝑎G𝑏G + 𝑎H𝑏H + 𝑎M𝑏M) + [ 𝑎H𝑏M − 𝑎M𝑏H 𝑖 + 𝑎M𝑏G − 𝑎G𝑏M 𝑗 + 𝑎G𝑏H − 𝑎H𝑏G 𝑘].
Firstofallwenoticethattheresultisnotapurevectorquaternionbecauseitcontainsascalar
part−(𝑎G𝑏G + 𝑎H𝑏H + 𝑎M𝑏M)thatwerecognizeas– (𝒂 ∙ 𝒃)(negativeofthedotproductof𝒂and
𝒃).Thevectorpart[insquarebrackets]isrecognizedasthecrossproduct𝒂×𝒃.Hencewehave
𝒂 ⋄ 𝒃 = 𝒂×𝒃 − 𝒂 ∙ 𝒃,
thatis,thequaternionproductofpurevectorquaternionsisequaltocrossproductminusdot
product.Wealsoseethatevenforpurevectorquaternionsthequaternionproduct𝒂 ⋄ 𝒃is
neithercommutativenoranticommutativealthoughitsfirstpart𝒂×𝒃isanticommutativeandthe
secondpart−𝒂 ∙ 𝒃iscommutative.Inspecialcases,however,itcanbeoneortheother.For
parallelvectors(𝒂×𝒃 = 𝟎)wehave𝒂 ⋄ 𝒃 = 𝒃 ⋄ 𝒂(commutativity)andfororthogonalvectors(𝒂 ∙
𝒃 = 0)wehave𝒂 ⋄ 𝒃 = −𝒃 ⋄ 𝒂(anticommutativity).
25Purevectorquaternionshavenoscalarpart(i.e.scalarpartiszero).TheycanbeconsideredasspacevectorsinℝMifwetaketheimaginaryunits𝑖, 𝑗, 𝑘asbasisvectors.(Infact,thebasisvectorsofℝMareoftendenotedby𝑖, 𝑗, 𝑘insteadofour𝒆G, 𝒆H, 𝒆M.)Theidentificationofpurevectorquaternionswithvectorsishereemphasizedbyusingsymbols𝒂and𝒃forthem.
33
Example.Usequaternionstofind(a)dotandcrossproductsofthevectors𝒂 = 2𝒆G + 𝒆H − 𝒆Mand
𝒃 = 5𝒆G − 3𝒆H + 6𝒆M.Usetheresultstofind(b)theanglebetweenthevectorsand(c)theareaof
theparallelogramdefinedbythevectors.
Solution:(a)Firstwritethevectorsaspurequaternions𝒂 = 2𝑖 + 𝑗 − 𝑘and𝒃 = 5𝑖 − 3𝑗 + 7𝑘.
Thenmultiplythemtermbytermtoget
𝒂 ⋄ 𝒃 = − 2 ∙ 5 + 1 ∙ −3 + −1 ∙ 6 + [ 1 ∙ 6 − −1 ∙ −3 𝑖 + −1 ∙ 5 − 2 ∙ 6 𝑗
+ 2 ∙ −3 − 1 ∙ 5 𝑘]
= −1 + [3𝑖 − 19𝑗 − 11𝑘].
Hence𝒂 ∙ 𝒃 = −1and𝒂×𝒃 = 3𝑖 − 19𝑗 − 11𝑘.
(b)Wehave 𝒂 = 2H + 1H + −1 H = 6and 𝒃 = 5H + −3 H + 7H = 83.Forthe
requiredangle𝜃wehavecos 𝜃 = (𝒂 ∙ 𝒃) ( 𝒂 ∙ 𝒃 ) = −1 498 fromwhichthecalculator
gives𝜃 ≈ 92.6°.
(c)Wehave 𝒂×𝒃 = 3H + −19 H + −11 H = 491 ≈ 22.2whichistherequestedarea.
∎
QUATERNIONSASAMAGICBOX
Wehaveseenthathiddenintheboxofquaternionsonecanfindallpreviouslyconsidered
versionsofvectormultiplication,atleastuptodimensionthree.Wejustfoundthatscalar(dot)
andvector(cross)productsarejustscalarandvectorpartsofthequaternionproduct.Alsothe
complex(rotational)productisaspecialcaseofquaternionproductsincecomplexnumbersare
quaternionsinwhich𝑗and𝑘donotappear.
Thepowerofquaternionsisnotlimitedtocombiningtheproductsdefinedearlier.Therearealso
newthingswecandowithquaternions:onesuchthingisrotationinspacewhichwewillconsider
below.FirstwewillgeneralizetheEulerequation(formulatedaboveforcomplexnumbers)to
quaternions.
34
EULEREQUATIONFORQUATERNIONS
RecalltheEulerequation𝑒o� = cos 𝜃 + 𝑖 sin 𝜃where𝑖isthecompleximaginaryunitand𝜃isa
realnumber(seep.17).Thisequationisequallytrueforquaternionsbecausecomplexnumbers
andtheimaginaryunit𝑖arealsoquaternionsforwhichthequaternionproductisthesameasthe
complexproduct.Itisalsotrueforthetwootherimaginaryunits𝑗and𝑘whichbehaveexactlyas𝑖
does,thatis,𝑖H = 𝑗H = 𝑘H = −1whichwasallthatwasneededtoprovetheEulerequation.
Hence𝑒p� = cos 𝜃 + 𝑗 sin 𝜃and𝑒�� = cos 𝜃 + 𝑘 sin 𝜃.Butisthatall?Canwefindmorequaternions𝑢withtheproperty𝑢H = −1?Theyallwould
automaticallysatisfyEulerequation.Soconsiderageneralpurevectorquaternion𝑢 = 𝑢G𝑖 + 𝑢H𝑗 + 𝑢M𝑘wherethecoefficients𝑢Jare
realnumbers.Asnotedabove,wecanidentify𝑢asavectorinℝM(eventhoughwedon’tuse
boldfacesymbolbecauseweconsider𝑢asa”number”)andasprovedabovewecanexpressthe
quaternionproduct𝑢H = 𝑢𝑢intermsofcrossanddotproductsasfollows
𝑢H = 𝑢𝑢 = 𝑢×𝑢 − 𝑢 ∙ 𝑢 [but𝑢×𝑢 = 0]
= −𝑢 ∙ 𝑢
= − 𝑢 H.
Weconcludethatthecondition𝑢H = −1holdstrueforallunitvectorsinℝMwhenthesevectors
aretreatedasquaternions.Geometricallytheseunitvectorssupporttheorigin-centredunit
sphereinℝM.Theimaginaryunits𝑖, 𝑗and𝑘arejustthreeexamplesofthose(infinitelymany)unit
vectors.Wehaveobtainedthefollowingresult:
GeneralizedEulerequationforquaternions
Foranyunitvector𝑢inℝMandarealnumber𝜃wehave𝑒�� = cos 𝜃 + 𝑢 sin 𝜃where𝑢isunderstood26asapurevectorquaternion.Thatis,allmultiplicationscontainedintheequationarequaternionproducts.
Inthenextsectionwe’llusethisresulttohandlespatialrotationsalgebraicallyandthusmake
themaptforcomputerswhichareveryefficientalgebraiccalculators(unlikewehumans).
26Thismeansthatallalgebraicoperations(especiallymultiplication)intheequationareinterpretedasoperationsforquaternions.
35
ROTATIONINTHE3-DIMENSIONALSPACE
Rotationinthe2D-planeisspecifiedbyanaxispoint(theorigin)andtherotationangle𝜃.We
foundabove(seeFigure10)thatthecomplexexpression𝑒o�𝑧givestheresultofrotatingthepoint
(orvector)𝑧byanangle𝜃abouttheorigin.Todefinearotationinthe3D-spacetheaxispoint
mustbereplacedbyanaxislinewhichisusuallyspecifiedbyanaxisvector.Thatis,werotatea
givenpoint(vector)𝑞byagivenangle𝜃aroundtheaxislinegivenbya3-Dvector𝑢.Withhelpof
thequaternionalgebrawecanexpresstherotatedpoint(vector)𝑞′intermsof𝑞, 𝑢and𝜃.
Tosimplifytheexpressionweassumethattheaxisvector𝑢isaunitvector.Wehave
Quaternionic”sandwich”rotationformulain3D-space
Ifapoint(vector)𝑞isrotatedbyanangle𝜃aroundanaxislinedeterminedbytheunitvector𝑢
(purevectorquaternion)thentherotatedpoint(vector)𝑞′isgivenbythequaternionicformula
𝑞� = 𝑒�(� H) ∙ 𝑞 ∙ 𝑒� �� H ,
whereallproductsarequaternionic(includingthosemarkedherebydotsforthesakeofclarity).
Thenickname”sandwich”referstothevisualstructureoftheformulawheretheoriginalpoint
(vector)𝑞is”sandwiched”betweenthetwoexponentialfactors.
Wewillcomebacktotheproofofthisformulalater.Atthisstageweonlytestthevalidityofthe
formulainasimplecasewheretheresultcanbeverifiedbyvisualintuition.
Example.Rotatethepoint𝑞 = 𝑎, 0,0 ,thatisthevector𝑞 = 𝑎𝑖,bytheangle𝜃 = 2𝜋 3(120°)
aroundtheaxisdeterminedbythevector𝑖 + 𝑗 + 𝑘.Findtherotatedpoint𝑞′.
Solution:Intheformulatheaxisvector𝑢mustbeaunitvector.Because 𝑖 + 𝑗 + 𝑘 = 3we
havetoput𝑢 = GM(𝑖 + 𝑗 + 𝑘).Wealsohave𝜃 2 = 𝜋 3.HencebythegeneralizedEulerequation
wehave
𝑒�(� H) = cos 𝜋 3 + 𝑢 sin 𝜋 3 =12 + 𝑢
32 =
12 1 + 𝑖 + 𝑗 + 𝑘 ,
36
𝑒�(�� H) = cos −𝜋 3 + 𝑢 sin −𝜋 3 =12 − 𝑢
32 =
12 1 − 𝑖 − 𝑗 − 𝑘 .
Substitutingtotheformulagivesfor𝑞�
𝑞� = 𝑒�(� H) ∙ 𝑞 ∙ 𝑒� �� H =12 1 + 𝑖 + 𝑗 + 𝑘 ∙ 𝑎𝑖 ∙
12 1 − 𝑖 − 𝑗 − 𝑘 .
Doingthequaternionmultiplicationscarefullystepbystepfromlefttorightgivesthen
𝑞′ =14 𝑎𝑖 − 𝑎 − 𝑎𝑘 + 𝑎𝑗 ∙ 1 − 𝑖 − 𝑗 − 𝑘
=14 𝑎 −1 + 𝑖 + 𝑗 − 𝑘 ∙ 1 − 𝑖 − 𝑗 − 𝑘
=14 𝑎 −1 + 𝑖 + 𝑗 + 𝑘 + 𝑖 + 1 − 𝑘 + 𝑗 + 𝑗 + 𝑘 + 1 − 𝑖 − 𝑘 + 𝑗 − 𝑖 − 1
=14 𝑎 4𝑗
= 𝑎𝑗
Soaccordingtotheformulathepoint𝑞 = 𝑎𝑖 = (𝑎, 0, 0)onthe𝑥-axishasbeenrotatedtothe
point𝑞� = 𝑎𝑗 = (0, 𝑎, 0)onthe𝑦-axis.[Forexample,𝑞 = (5,0,0)goesto𝑞� = (0,5,0).]
Geometricintuitiontellsthatthisiscorrectsincerotationby120°(onethirdofthefullround)
aroundtheaxis𝑢whichsymmetricwithrespecttothecoordinateaxesturns𝑥-axistotheplaceof
𝑦-axis,𝑦-axistotheplaceof𝑧-axisandfinally𝑧-axistotheplaceof𝑥-axis.Thisexamplegivessomeconfidencetothevalidityoftheformulaalthoughitisnotaproof.
Figure12.Rotationby120°abouttheaxisvector𝑢 = 𝑖 + 𝑗 + 𝑘takesthepoint𝑞 = 𝑎𝑖 = (𝑎, 0, 0)onthe𝑥-axistothepoint𝑞′ = 𝑎𝑗 = (0, 𝑎, 0)asestablishedbythequaternionrotationformula.
37
NOTE:Thequaternionicrotationformulafor3D-spacewasfoundalreadybyHamiltoninca.1845.Therequiredtedious(thoughroutine)calculationsrenderedtheuseofthisformulaquiteimpracticalformorethanacenturyuntiltheadventofcomputerswhichcandothealgebrainalmostnotime.Nowthisformulahasbecomeanessentialtoolincomputergraphics,aeronauticsandspacenavigation.
WHYNOTSTOPHERE?
Quaternionproductseemsanidealgeneralizationofthepreviousproducts(scalar,complex,
vector)whichareallincludedaspartsofthequaternionproduct.Italsoobeystheimportantlaws
ofassociativityanddistributivityalthoughcommutativityandanticommutativitybothfailexceptin
specialcases.However,themostseriouslimitationofthequaternionproductisthatitworksonly
inthe4D-worldℝY(orℝ×ℝMifyouprefertoconsiderquaternionproductasanoperationwith
thecombinationsscalarsand3D-vectors).Thereforewecontinuediggingdeeper!
38
8.OUTERPRODUCTORWEDGEPRODUCT𝒂 ∧ 𝒃
Inchapter6westudiedthevector(cross)product𝒂×𝒃which,bydefinition,isavector
perpendicularto𝒂and𝒃(intheright-handsense)anditsabsolutevalueisequaltotheareaof
theparallelogramenclosedbythevectors𝒂and𝒃.
Figure13.Theabsolutevalue(norm,length)ofthevector𝒂×𝒃isequaltotheareaoftheparallelogramenclosedbythevectors𝒂and𝒃.Bytrigonometrywehave 𝒂×𝒃 = 𝒂 𝒃 sin 𝜃.Thevector𝒂×𝒃itselfisperpendicularto𝒂and𝒃andformsaright-handsystemwiththem.
Intheendofchapter6wenoticedonedisadvantageinthenotionofthecrossproduct:its
direction(perpendiculartoitsfactors)isonlywell-definedinℝM.Ournextcandidateforthevector
multiplication–theouterproduct–avoidsthisproblemwhilepreservesitsabsolutevalue.Itwas
Grassmann’sidea(ca.1845)toallownew”inhabitants”tothekingdomofvectors.Onenewtribe
isgeneratedbytheouterproducts𝒂 ∧ 𝒃.Theresultofanouterproductisnotavectorbuta
bivector,anordered(ororiented)pairofvectors.Alogicalcontinuationofthisliberal
”immigrationpolicy”isthentoadmitalsotrivectors(3-vectors),4-vectorsandingeneral𝑛-vectors
toourdomainofdiscourse.Besidesthemitturnsoutusefultogrant”fullcitizenship”toscalars
whichsofarhaveplayedasiderole.AllthiswillfinallyexpandourfamiliarvectorspacesℝJinto
”multiethnic”spacesofmultivectors𝔾J.Butletusnowstartthisprocessbytheintroductionof
bivectorsasouterproductsoftraditionalvectors.
39
Definitionoftheouterproduct𝒂 ∧ 𝒃
Figure14.Theouterproduct(orwedge27product)ofthevectors𝒂and𝒃isthebivector𝒂 ∧ 𝒃.Itsabsolutevalue(norm)isequaltotheareaoftheparallelogramenclosedbythevectors𝒂and𝒃.Wehave 𝒂 ∧ 𝒃 = 𝒂 𝒃 sin 𝜃 .Thebivectors𝒂 ∧ 𝒃and𝒃 ∧ 𝒂havethesameabsolutevaluebutoppositeorientations.Thebivector𝒂 ∧ 𝒃hasapositive(anticlockwise)orientation𝒂,𝒃,−𝒂,−𝒃.Thebivectors𝒃 ∧ 𝒂hasanegative(clockwise)orientation𝒃, 𝒂, −𝒃,−𝒂.Hence𝒂 ∧ 𝒃 = −𝒃 ∧ 𝒂.Theouterproductbehavesalgebraicallyexactlylikethecrossproduct.Thecrucialdifferenceisgeometric.Thecrossproduct𝒂×𝒃isavector,aone-dimensionallinearobject.Theouterproduct𝒂 ∧ 𝒃isabivector,atwo-dimensionalplanarobject.
Youseethattheouterproduct𝒂 ∧ 𝒃andthevectorproduct𝒂×𝒃areverycloserelatives.The
greatadvantageoftheformeristhatitgeneralizeseasilytohigherdimensionalspacesℝJ.Thisis
sobecause𝒂 ∧ 𝒃isanobjectinaplanewhichiswell-definedinalldimensionswhile𝒂×𝒃isan
object(vector)perpendiculartothatplaneandinhigherdimensions(𝑛 ≥ 4)theperpendicular
directionisnotuniquelydefinedasitisinℝM.
Propertiesoftheouterproduct
(1)If𝒂and𝒃areparallel,then 𝒂 ∧ 𝒃 = 0.Inparticular 𝒂 ∧ 𝒂 = 0.
(2)If𝒂and𝒃areperpendicular,then 𝒂 ∧ 𝒃 = 𝒂 𝒃 .
(3)𝒂 ∧ 𝒃 = −𝒃 ∧ 𝒂 [anticommutativity]
(4)𝑡 𝒂 ∧ 𝒃 = 𝑡𝒂 ∧ 𝒃 = 𝒂 ∧ (𝑡𝒃) [freemobilityofscalarfactor]
(5)𝒂 ∧ 𝒃 + 𝒄 = 𝒂 ∧ 𝒃 + 𝒂 ∧ 𝒄 [distributivityoveraddition]
Theaboveproperties(1)–(5)canbeverifiedusingtheparallelogramvisualization.In(1)wewrite
simplythat𝒂 ∧ 𝒃 = 0.Whenoperatingwithmultivectorsitturnsoutusefultoidentifydifferent
27Theword”wedge”isderivedfromthesymbol∧.
40
kindsofzeros(scalarzero,vectorzero,bivectorzero,…)asoneandsinglezero(denotedby0)of
themultivectorspace.
Notethatintheabovedefinitionoftheouterproductweareusingparallelogramasageometric
illustrationoftheproduct𝒂 ∧ 𝒃.Thisshouldnotbetakentooliterally.Theouterproductdoesnot
haveanyspecific”shape”.Essentiallytheouterproduct𝒂 ∧ 𝒃representsgeometricallyan
orientedplanedefinedbyvectors𝒂and𝒃andwithanattachedabsolutevalue(norm)
𝒂 𝒃 sin 𝜃.Changingtheorderofvectorsreversestheorientation(fromplustominusorvice
versa)oftheplane.Ifyouchoosetocall𝒂 ∧ 𝒃positiveorderthen𝒃 ∧ 𝒂isnegative.Thereisno
”natural”or”absolute”ruleforthesign.Itisamatterofagreementbutyouhavetofollowthe
logicalconsequencesofyourinitialagreement.Thingsarepositiveornegativeinrelationtoeach
other.
Thatsaid,weshouldnotforgetthatmoreconcretevisualizationsfor𝒂 ∧ 𝒃(likeanoriented
parallelogramororientedcircle)areoftenveryhelpful.
OUTERPRODUCTOFMULTIPLEVECTORSTheparallelogramvisualizationprovidesanaturalwaytogeneralizetheouterproductforthreeor
morevectors.Theproduct𝒂 ∧ 𝒃 ∧ 𝒄isdefinedasanorientedparallelpiped(oftencalledjust3D-
parallelogram,seeFigure15below).Thenorm 𝒂 ∧ 𝒃 ∧ 𝒄 isequaltothevolumeofthissolid.Its
orientation(+or–)isdefinedbycomparingtheorderofvectorswithsomechosenorderlabelled
aspositive.Soifwechoose𝒂 ∧ 𝒃 ∧ 𝒄asabasicpositiveorderthen𝒃 ∧ 𝒂 ∧ 𝒄,𝒂 ∧ 𝒄 ∧ 𝒃and𝒄 ∧
𝒃 ∧ 𝒂havenegativeorientation.Hence,forexample,𝒃 ∧ 𝒂 ∧ 𝒄 = −𝒂 ∧ 𝒃 ∧ 𝒄.Ontheotherhand
𝒂 ∧ 𝒃 ∧ 𝒄aswellas𝒃 ∧ 𝒄 ∧ 𝒂and𝒄 ∧ 𝒂 ∧ 𝒃areallequalhavingapositiveorientation.Theruleis
thatswappingtwoadjacentvectorschangesthesignoftheproduct.(Henceincaseofthree
vectorswehave6 = 3 ∙ 2 ∙ 1permutationsofwhichthreearepositiveandthreenegative.Incase
offourvectorsthereare24permutations,12positiveand12negative.Thesamelogicworksfor
anynumberofvectors.)
41
Figure15.Theleftsolid(3D-parallelogramorparallelpiped)illustratestheouterproduct𝒂 ∧ 𝒃 ∧ 𝒄whichiscalledatrivector28.Itsnorm 𝒂 ∧ 𝒃 ∧ 𝒄 isequaltothevolumeofthesolid.Itisalwayspossibletodeformthe3D-parallelogramintoaright-angledbox(ontheright)bykeepingtheside𝒂andthedistancesbetweenoppositefacesfixed.Inthisprocess29thesidevectors𝒃and𝒄willshortentovectors𝒃′and𝒄′(whichmakerightangleswith𝒂)butthevolumeofthesolidremainsconstant.Then𝒂 ∧ 𝒃′ ∧ 𝒄′ = 𝒂 ∧ 𝒃 ∧ 𝒄becausetheyhavethesameorientationandnorm(volume).Sowehave 𝒂 ∧ 𝒃 ∧ 𝒄 = 𝒂 ∧ 𝒃′ ∧ 𝒄′ = 𝒂 𝒃′ 𝒄′ sincethevolumeofaboxisobtainedbymultiplyingthelenghtsoftheperpendicularsides.
NOTE:Noticethatifanytwovectorsintheproduct𝒂 ∧ 𝒃 ∧ 𝒄areparallelthenallthreevectorsareinthesameplaneandtheparallelpipedbecomesaflatplanefigurewithzerovolume.(So,forexample, 𝒂 ∧ 𝒃 ∧ 𝒂 = 0andwewrite𝒂 ∧ 𝒃 ∧ 𝒂 = 𝟎.)Thesameistrueifoneofthevectorsisalinearcombinationoftwoothers(say𝒄 = 2𝒂 − 7𝒃).Againtheparallelpipedflattensontotheplaneofthevectors𝒂and𝒃whichmakestheproduct𝒂 ∧ 𝒃 ∧ 𝒄vanish.
ASSOCIATIVITYOFTHEOUTERPRODUCT
Theaboveconsiderationsshowthatwecanworkwithouterproductsofarbitrarilymanyvectors.
Italsoopensthepossibilitytotalkaboutouterproductsof,say,avector𝒂andabivector𝒃 ∧ 𝒄.
Theobviouswaytodefinetheproducts𝒂 ∧ (𝒃 ∧ 𝒄)and(𝒂 ∧ 𝒃 ∧ 𝒄is,ofcourse,toset
(6)𝒂 ∧ 𝒃 ∧ 𝒄 = 𝒂 ∧ 𝒃 ∧ 𝒄 = (𝒂 ∧ 𝒃) ∧ 𝒄,
whichalsoshowsthatouterproductsatisfiestheruleofassociativityandweadditasthesixth
propertyofouterproduct.Youarefreetocarryouttheouterproductstepbystepmultiplying
subgroupsasyoufindconvenient.Youarealsofreetochangetheorderoffactorsbyswapping
28Rememberthat𝒂 ∧ 𝒃wascalledabivector(or2-vector)Analogously𝒂 ∧ 𝒃 ∧ 𝒄iscalledtrivector(or3-vector).Ingeneraltheouterproductsof𝑛vectorsarecalled𝑛-vectors.29ThereisastandardalgebraicprocedurecalledGram-Schmidtorthogonalizationtoconstructthisequivalentright-angledboxfromagivenparallelpiped.TheG-S-procedureworksinalldimensions.
42
adjacentvectorsbutremembertochangethesign.Everyswapchangestheorientation.Ifyou
startwithapositiveorderthenanevennumberofswapsleadstoapositiveandanoddnumber
leadstoanegativeorder.
Wewillnowtakeacloserlookatthenewperspectivesthattheouterproductopensforvector
algebraindifferentdimensions.Wewillmeetanumberofnewanimalsthatinvadeoursofar
vector-dominatedworldfromtheloopholeopenedbytheouterproduct.
OUTERPRODUCTSINℝHANDIN𝔾H
Letusfirstconsider2D-vectorsinℝH.Assume𝒆Gand𝒆Haretheusualorthonormalbasisvectors
forℝH.Bythedefinitionofouterproductwehave:
𝒆G ∧ 𝒆G = 𝒆H ∧ 𝒆H = 0, [parallelvectorsproducezero]
𝒆G ∧ 𝒆H = −𝒆H ∧ 𝒆G. [byanticommutativityof∧]
Noticeinthelattercasethat𝒆G ∧ 𝒆Hand𝒆H ∧ 𝒆Garebivectorsandcannotbesimplifiedany
further.Weonlynowthattheyareoppositetoeachother.Wecan,ofcourse,calculatetheir
norms: 𝒆G ∧ 𝒆H = 𝒆H ∧ 𝒆G = 𝒆G 𝒆H sin 90° = 1.Ifnow𝒂and𝒃arearbitraryvectorsinℝHthen𝒂 = 𝑎G𝒆G + 𝑎H𝒆Hand𝒃 = 𝑏G𝒆G + 𝑏H𝒆H,where
𝑎�, 𝑏�areappropriatescalarcoefficients(thecoordinatesof𝒂and𝒃).Bythedistributivelawwe
havenow
𝒂 ∧ 𝒃 = 𝑎G𝒆G + 𝑎H𝒆H ∧ 𝑏G𝒆G + 𝑏H𝒆H
= 𝑎G𝑏G 𝒆G ∧ 𝒆G + 𝑎G𝑏H 𝒆G ∧ 𝒆H + 𝑎H𝑏G 𝒆H ∧ 𝒆G + 𝑎H𝑏H 𝒆H ∧ 𝒆H
= 𝑎G𝑏H 𝒆G ∧ 𝒆H − 𝑎H𝑏G 𝒆G ∧ 𝒆H
= 𝑎G𝑏H − 𝑎H𝑏G 𝒆G ∧ 𝒆H
= 𝑎G𝑏H − 𝑎H𝑏G 𝒆G ∧ 𝒆H.
Thenorm 𝒂 ∧ 𝒃 = 𝑎G𝑏H − 𝑎H𝑏G because 𝒆G ∧ 𝒆H = 1.
YouseethatifyouwanttodoouterproductsofvectorsinℝHwehavetoacceptanewcitizen,
namelythebasicbivector𝒆G ∧ 𝒆H,tocomplementthebasisvectors𝒆Gand𝒆H.Ithasturnedout
usefulalsotoincorporaterealnumbers(scalars)asestablishedcitizensofourtwo-dimensional
43
kingdomwhichwenowdesignatewiththesymbol𝔾H.Wewantournewkingdom𝔾Hbeclosed
underaddition.Hencewehavetoacceptallsums𝑆 = 𝑡 + 𝒂 + 𝒄 ∧ 𝒅(scalar+vector+bivector)as
legitimatecitizensof𝔾H.Itisaremarkablefactthisextensionstillkeeps𝔾Hclosedunderouter
product.Namelyifweout-multiplytwosuchsums𝑆G = 𝑡G + 𝒂G + 𝒄G ∧ 𝒅Gand𝑆H = 𝑡H + 𝒂H +
𝒄H ∧ 𝒅Hweget(assumingthatouterproductwithscalarsisjustusualscalarmultiplication)
𝑆G ∧ 𝑆H = 𝑡G + 𝒂G + 𝒄G ∧ 𝒅G ∧ 𝑡H + 𝒂H + 𝒄H ∧ 𝒅H
= 𝑡G𝑡H + 𝑡G𝒂H + 𝑡H𝒂G + 𝒂G ∧ 𝒂H + 𝒂H ∧ 𝒂G + 𝒂G ∧ 𝒄H ∧ 𝒅H + 𝒄G ∧ 𝒅G ∧ 𝒂H
+ 𝒄G ∧ 𝒅G ∧ 𝒄H ∧ 𝒅H ,
butnoticingthatbyanticommutativity𝒂G ∧ 𝒂H + 𝒂H ∧ 𝒂G = 𝟎wehave
𝑆G ∧ 𝑆H = 𝑡G𝑡H + 𝑡G𝒂H + 𝑡H𝒂G + 𝒂G ∧ 𝒄H ∧ 𝒅H + 𝒄G ∧ 𝒅G ∧ 𝒂H + 𝒄G ∧ 𝒅G ∧ 𝒄H ∧ 𝒅H
wherewedeletedthebracketsbytheassociativityofouterproduct.Butheretheproductsof
threefactorsmustbezerobecausethreeℝH-vectorsareinthesameplane(namelyinℝH).Butso
isthelastproductoffourvectors,too,since–byassociativity–wecantakethegroupofthefirst
threefactors𝒄G ∧ 𝒅G ∧ 𝒄Hwhichmustbezeroandconsequentlythewholeproductiszero.Sothe
outerproductoftwoarbitrarycombinations𝑆Gand𝑆Hfinallyreducesto
𝑆G ∧ 𝑆H = 𝑡G𝑡H + 𝑡G𝒂H + 𝑡H𝒂G + 𝟎 ∧ 𝒅(scalar+vector+bivector)
whichisoftherequiredform(bivectorpartisjustzero).
So𝔾Htakenasasetofallsuchsums(scalar+ℝH-vector+ℝH-bivector)constitutesanalgebraically
closeddomainofobjects(calledmultivectors).Algebraicallyclosedmeans:closedunderthe
chosenoperations(herethechosenoperationsareaddition,scalarmultiplicationandouter
product).
Thedifferencebetweenvectoralgebraandgeometricalgebraisthattheformeroperateswith
objectsinℝH(vectors)whilethelatteroperateswithobjectsinalargerdomain𝔾H(multivectors).
BecauseℝHisjustapartof𝔾H(insymbolsℝH ⊂ 𝔾H)thevectoralgebraisjustapartofmore
generalgeometricalgebra.
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BASISFOR𝔾H
Thesetoftwoelements 𝒆G, 𝒆H makesabasisforℝH.Everyvector𝒂 ∈ ℝHcanbeexpressedasa
linearcombinationofthebasisvectors,𝒂 = 𝛼G𝒆G + 𝛼H𝒆H,where𝛼�arescalarcoefficients.
Analogouslywehaveabasisoffourelements 1, 𝒆G, 𝒆H, 𝒆G ∧ 𝒆H for𝔾H,themultivector
extensionofℝH.Thisissobecauseevery𝔾H-multivector𝐴 = 𝑟 + 𝑉 + 𝐵canbeexpressed30asa
linearcombinationofthesefourbasiselements,𝐴 = 𝛼� ∙ 1 + (𝛼G𝒆G + 𝛼H𝒆H) + 𝛼GH 𝒆G ∧ 𝒆H ,
whereagain𝛼�arescalarcoefficients.Thefirstpart𝛼� ∙ 1obviouslycatchesallrealnumbers𝑟,
thesecondpart𝛼G𝒆G + 𝛼H𝒆Hcatchesall2D-vectors𝑉andfinally𝛼GH 𝒆G ∧ 𝒆H catchesall
bivectors𝐵asshownatthebeginningofthepreviouschapter.Hencewemaycharacterize𝔾Hasa
four-dimensionaldomainbecauseeveryelementof𝔾Hcanbeidentifiedbyfourrealnumbers
𝛼�, 𝛼G, 𝛼Hand𝛼GH.
OUTERPRODUCTSINℝMANDIN𝔾M
Nowconsider3D-vectorsinℝM.Againlet 𝒆G, 𝒆H, 𝒆M betheusualorthonormalsetofbasisvectors
forℝM.AsaboveincaseofℝHwehaveforindices𝑛,𝑚 = 1, 2, 3thefollowingbivectors:
𝒆J ∧ 𝒆J = 𝟎, [parallelvectorsproducezero]
𝒆¡ ∧ 𝒆J = −𝒆J ∧ 𝒆¡for𝑚 ≠ 𝑛. [byanticommutativityof∧]
Butnowtherearealsonon-zerotrivectors,namely𝒆G ∧ 𝒆H ∧ 𝒆M = 𝒆H ∧ 𝒆M ∧ 𝒆G = 𝒆M ∧ 𝒆H ∧ 𝒆G
whichhavethesameorientationaswellas𝒆G ∧ 𝒆M ∧ 𝒆H = 𝒆H ∧ 𝒆G ∧ 𝒆M = 𝒆M ∧ 𝒆G ∧ 𝒆Hwhich
havetheoppositeorientationsothat,forexample,𝒆H ∧ 𝒆G ∧ 𝒆M = −𝒆G ∧ 𝒆H ∧ 𝒆M.Allthese
trivectorshavethesamenormwhich(asexplainedinFigure15above)mustbeequalto1because
thecorresponding3D-parallelogramistheunitcube.Othertrivectorslike𝒆G ∧ 𝒆H ∧ 𝒆Gwherethe
samefactorisrepeatedare,ofcourse,zeros(volume=0).So,essentially,thereisonlyonenon-
zerotrivectorinℝM(ortobemoreprecise,in𝔾M).Again,inanalogywith𝔾H,therearenononzero
productsoftheform𝒂 ∧ 𝒃 ∧ 𝒄 ∧ 𝒅in𝔾M.
30Inthisexpressionof𝑀wedenoteitsscalarpartby𝑟,vectorpartby𝑉andbivectorpartby𝐵.Every𝔾H-multivectorconsistsofthesethreeparts(someofthemcanbezeros)sothatscalars,vectorsandbivectorsarealsomultivectorsintheirownright.Multivectorsofhigherdimensionscanhavemoreparts:trivectors,4-vectors,etc.Notethatwhileweuseboldfaceitalicsymbols(like𝒂)forvectors,weuseonlyitalicsformultivectorsandtheirparts(like𝑀,𝑉, 𝐵).
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AsaboveincaseofℝHand𝔾Hwecanexpresseverybivector𝒂 ∧ 𝒃asacombinationofthethree
basicbivectorsof𝔾M.So𝒂 ∧ 𝒃 = 𝛼GH 𝒆G ∧ 𝒆H + 𝛼GM 𝒆G ∧ 𝒆M + 𝛼HM 𝒆H ∧ 𝒆M ,where𝛼’sare
appropriatescalarcoefficients31.Likewiseeverytrivector𝒂 ∧ 𝒃 ∧ 𝒄isascalarmultipleofthe(only
one)basictrivector,thatis𝒂 ∧ 𝒃 ∧ 𝒄 = 𝛼GHM 𝒆G ∧ 𝒆H ∧ 𝒆M .Thesefactsfollow(asabovefor𝔾H)by
astraightforwardcalculationoftherespectiveouterproductswith𝒂, 𝒃and𝒄replacedbytheir
representationsintheℝM-basis 𝒆G, 𝒆H, 𝒆M .
Analogouslythe𝔾M-multivectorsconsistofsums:scalar+ℝM-vector+ℝM-bivector+ℝM-trivector.
Thebasisfor𝔾Misthentheset 1, 𝒆G, 𝒆H, 𝒆M, 𝒆G ∧ 𝒆H, 𝒆G ∧ 𝒆M, 𝒆H ∧ 𝒆M, 𝒆G ∧ 𝒆H ∧ 𝒆M which
containsonescalar(1),threevectors,threebivectorsandonetrivector,altogethereight
elements.Asyousee,thescalarelement1andthetrivectorelement𝒆G ∧ 𝒆H ∧ 𝒆Mareina
symmetricpositionwhichiswhythelatterissometimescalledapseudoscalarof𝔾M.Wewillsee
laterthatingeometricalgebrathepseudoscalarbehavesinmanywayslikeascalar.
OUTERPRODUCTSINℝJANDIN𝔾J
Itisnowprettyobvioushowthingsgeneralizetohigherthan3dimensions.Startingfroman
orthonormalbasis 𝒆G, 𝒆H, … , 𝒆J forℝJweproceedanalogouslytoconstructabasisfor𝔾J-
multivectors.Thisbasisconsistsof2Jelements:onescalar(1),𝑛vectors,𝑛(𝑛 − 1) 2bivectors,
𝑛(𝑛 − 1)(𝑛 − 2) 3!trivectors,...andfinallyonepseudoscalar𝒆G ∧ 𝒆H ∧ …∧ 𝒆J.
Inthesenoteswewillfocusontwo-andthree-dimensionalgeometricalgebras𝔾Hand𝔾M
althoughwewilloccasionallymakeobservationsonhigherdimensions,too.
CLOSUREOF𝔾JUNDERTHESCALARPRODUCTOFVECTORS
Itisworthnoticingthat𝔾H,thespaceof2D-multivectors𝑀 = 𝛼� + 𝛼G𝒆G + 𝛼H𝒆H + 𝛼GH(𝒆G ∧ 𝒆H)
isclosedunderscalar(dot)productofvectors.If𝒂and𝒃are2D-vectors(i.e.multivectorsfor
which𝛼� = 𝛼GH = 0),thentheirusualscalarproduct𝒂 ∙ 𝒃isarealnumberwhichisamemberof
𝔾H(asamultivectorforwhich𝛼G = 𝛼H = 𝛼GH = 0).Thesameistruegenerallyfor𝔾Jwhichare
thereforeclosedunderthescalarproductofvectors.Wecannotyetsaythat𝔾J,asawhole,is
31Thedoubleortripleindicesinthescalarcoefficient(𝛼GM𝑜𝑟𝛼GHM)refertotherespectivebasisbivector𝒆G ∧ 𝒆Mor𝒆G ∧ 𝒆H ∧ 𝒆M.
46
closedunderscalarproductbecausewehavenotdefinedwhatwemeanbyascalarproductof
multivectors.Thisquestionwillbetackledwhengeometricalgebraisdevelopedfurther(inPart2
ofthesenotes).
WHYNOTSTOPHERE?
Inthischapterwehaveintroducedstillanotherversionofmultiplicationforvectors,theouter
product𝒂 ∧ 𝒃.Itwasintendedtodothejobofvector(cross)product𝒂×𝒃whilebeingfreeofits
limitations(𝒂×𝒃onlyworkswellinℝM).Thiswasachievedbutwehadtopayapricebyallowing
newinhabitants(multivectors)torushintoourwell-regulateddomainofvectors(ℝJ),whichwas
therebyextendedtoamuchmoremulticulturaldomainofmultivectors(𝔾J).Soitseemsthatthe
wholepictureofvectoralgebrahasturnedmorechaotic.Butasitoftenhappens,something
radicallynewcancomeoutofchaos.AndthisindeedhappenedinastrokeofgeniuswhenWilliam
KingdonClifford(1845-1879)inventedtheconceptofgeometricproducttobringorderintothe
chaosofmultivectors.Soouradventuregoesonintoevenbroaderavenuesofgeometricalgebra.
47
48
9.GEOMETRICPRODUCTANDGEOMETRICALGEBRA
Clifforddefinedthegeometricproduct𝒂𝒃oftwovectors𝒂and𝒃asasumofinner(scalar,dot)
andouter(wedge)productbysetting
𝒂𝒃 = 𝒂 ∙ 𝒃 + 𝒂 ∧ 𝒃 [definitionofgeometricproductforvectors]
whichisavaliddefinitionforvectorsofalldimensions,thatisfor𝒂, 𝒃 ∈ ℝJwhere𝑛isany
positiveinteger.Thisistruebecause(aswehaveseen)theinnerandouterproductsaredefinedin
alldimensions.
Thegeometricproductisdenotedsimplyby𝒂𝒃whichisanappropriatenotationbecauseofthe
fundamentalnatureofthisproduct.Noticethatdespiteitssuggestivenamethegeometric
productispurelyanalgebraicconstruct.Thereisnonaturalvisualillustrationforthisproductin
thesamesenseasfortheouterproduct(parallelogram)orscalarproduct(directedprojection).
Example:Forvectors𝒂 = 2𝒆G − 𝒆Hand𝒃 = 3𝒆G + 2𝒆Hfindthegeometricproducts𝒂𝒃and𝒃𝒂.
Solution:Wehave
𝒂 ∙ 𝒃 = 2 ∙ 3 + −1 ∙ 2 = 4,
𝒂 ∧ 𝒃 = 6𝒆G ∧ 𝒆G + 4𝒆G ∧ 𝒆H − 3𝒆H ∧ 𝒆G − 2𝒆H ∧ 𝒆H
= 4𝒆G ∧ 𝒆H + 3𝒆G ∧ 𝒆H
= 7𝒆G ∧ 𝒆H,
andhence
𝒂𝒃 = 𝒂 ∙ 𝒃 + 𝒂 ∧ 𝒃 = 4 + 7(𝒆G ∧ 𝒆H)
and
𝒃𝒂 = 𝒃 ∙ 𝒂 + 𝒃 ∧ 𝒂 = 𝒂 ∙ 𝒃 − 𝒂 ∧ 𝒃 = 4 − 7(𝒆G ∧ 𝒆H).
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NOTE:Weseethatthegeometricmultiplicationproducesmultivectorseventhoughweworkwithvectors.ThegeometricproductleadsautomaticallyoutofℝH(orℝJingeneral)into𝔾H(or𝔾Jingeneral).Propertiesofgeometricproduct
Foranyvectors𝒂, 𝒃, 𝒄 ∈ ℝJandscalar𝛼thefollowingaretrue:
(1)𝒂𝒃 ∈ 𝔾J. [closurein𝔾J]
(2a)𝒂 𝒃 + 𝒄 = 𝒂𝒃 + 𝒂𝒄 [distributivityfromtheleft]
(2b) 𝒃 + 𝒄 𝒂 = 𝒃𝒂 + 𝒄𝒂 [distributivityfromtheright]
(3)(𝛼𝒂)𝒃 = 𝒂 𝛼𝒃 = 𝛼(𝒂𝒃) [freemobilityofascalarfactor]
(4) 𝒂𝒃 𝒄 = 𝒂(𝒃𝒄) [associativity]
(5)1𝒂 = 𝒂1 = 𝒂 [1isaneutralmultiplier]
(6a)𝒂𝒃 = 𝒂 ∙ 𝒃 = 𝒂 𝒃 [ifthevectors𝒂and𝒃areparallel]
(6b)𝒂𝒃 = 𝒃𝒂 [ifthevectors𝒂and𝒃areparallel]
(6c)𝒂𝒂 = 𝒂 ∙ 𝒂 = 𝒂 𝟐 [geometricsquareisthevector’snormsquared]
(7a)𝒂𝒃 = 𝒂 ∧ 𝒃 [ifvectors𝒂and𝒃areorthogonal]
(7b)𝒂𝒃 = −𝒃𝒂 [ifvectors𝒂and𝒃areorthogonal]
(8)𝒂𝒃 ≠ 𝒃𝒂and𝒂𝒃 ≠ −𝒃𝒂 [ifvectors𝒂and𝒃areneitherparallelnororthogonal]
Proof:Asanexample,weprovethelaw(2a).
Let𝒂, 𝒃, 𝒄bearbitraryvectorsinℝJ.Thenwehave
𝒂 𝒃 + 𝒄 = 𝒂 ∙ 𝒃 + 𝒄 + 𝒂 ∧ 𝒃 + 𝒄 [bydefinitionofgeometricproduct]
= 𝒂 ∙ 𝒃 + 𝒂 ∙ 𝒄 + 𝒂 ∧ 𝒃 + 𝒂 ∧ 𝒄 [bydistributivityofinnerandouterprod.]
= (𝒂 ∙ 𝒃 + 𝒂 ∧ 𝒃) + (𝒂 ∙ 𝒄 + 𝒂 ∧ 𝒄)
= 𝒂𝒃 + 𝒂𝒄. [bydefinitionofgeometricproduct]
Proofsofotherrules(excepttherule(4)ofassociativity)areequallystraightforward.Toproverule
(4)weneedtodefinethegeometricproductnotonlyforvectorsbutformultivectorsingeneral.
Thiswillbedonelater(inPart2).Forthetimebeingwetakeitforgrantedthattheassociativity
holdstrue.
50
NOTE:Noticetheproperty(8)thatthegeometricproductdoesnotsatisfyanygeneralruleofcommutativitynoranticommutativity.Thisisaninevitableconsequenceofitsdefinitionasasumofscalarproduct(commutative)andwedgeproduct(anticommutative).Itmightseemaproblemthatthisusefulalgebraicruleisnowunavailable.However,asnotedin(6a)-(6c),forparallelvectorswehavecommutativitysincethewedgeproductdisappears.Respectively,asnotedin(7),fororthogonalvectorswehaveanticommutativitysincethescalarproductdisappears.Itturnsoutthatthesetwospecialruleslargelycompensatethelossofageneralrule.
GEOMETRICPRODUCTSOFORTHONORMALBASISVECTORS
Assumethat 𝒆G, 𝒆H, … , 𝒆J�G, 𝒆J isasetoforthonormalbasisvectorsforℝJ.Byrules(6c)and
(7a)theirmutualgeometricproductsare
𝒆�𝒆� = 1, [scalar]
𝒆�𝒆¤ = 𝒆� ∧ 𝒆¤ for𝑘 ≠ 𝑙 [bivector]
Forfor𝑘 ≠ 𝑙wehave𝒆�𝒆¤ = 𝒆¤𝒆�.Weseethat𝔾Jhas𝑛(𝑛 − 1)/2independentbasicbivectors
𝒆�𝒆¤.Besidesthem32thebasisof𝔾Jcontainstrivectors𝒆p𝒆�𝒆¤,4-vectors33etc.andfinallythe
(onlyone)𝑛-vector𝒆G𝒆H …𝒆J.(Differentpermutationsofthisproductareallequaloroppositeto
eachother.)Andagain,ifyouformaproductofmorethan𝑛basisvectors𝒆�thensuchaproduct
willreduce,byrules(6)and(7),intoaproductofgrade𝑛orlower.Hencethetotalnumberof
independentbasiselementsin𝔾Jis2J.
GEOMETRICALGEBRAIN𝔾H
Let’stakeacloserlookatthedomainℝHoftwo-dimensionalvectorswiththeorthonormalbasis
𝒆G, 𝒆H anditsmultivectorextension𝔾Hwiththeextendedbasisbasis 1, 𝒆G, 𝒆H, 𝒆G𝒆H offour
elements.Wehavealreadyseenabovethat𝔾Hisclosedundertheoperationsofaddition,scalar
multiplicationandouterproduct.Nowwecomputethegeometricproductoftwo𝔾H-multivectors 𝐴 = 𝛼� ∙ 1 + 𝛼G𝒆G + 𝛼H𝒆H + 𝛼GH 𝒆G𝒆H ,
𝐵 = 𝛽� ∙ 1 + (𝛽G𝒆G + 𝛽H𝒆H) + 𝛽GH 𝒆G𝒆H .
32Bivectorsweredefinedasouterproducts𝒆� ∧ 𝒆¤ butwecanalsowritethemasgeometricproducts𝒆�𝒆¤ sincefororthogonalvectorsthetwoproductsareidentical.Thisisnottruefornon-orthogonalvectors.33Inmoregeneralterminologyvector=1-vector,bivector=2-vector,trivector=3-vectoretc.Notethatthe4-vectorsofgeometricalgebrahavenothingtodowiththe4-vectorsofrelativitytheory.
51
Asaninitialstepwecalculatethegeometricproductofthebivectors 𝒆G𝒆H 𝒆G𝒆H .Wehave
𝒆G𝒆H 𝒆G𝒆H = 𝒆G(𝒆H𝒆G)𝒆H [byassociativityrule(4)]
= −𝒆G(𝒆G𝒆H)𝒆H [byortho-anti-comm.rule(7b)]
= −(𝒆G𝒆G)(𝒆H𝒆H) [byassociativityrule(4)]
= −(1)(1) [by(6c)]
= −1.
Sowehaveaninterestingresult 𝒆G𝒆H H = 𝒆G𝒆H 𝒆G𝒆H = −1,somethingwehaveseenearlier
inanothercontext(andwewillmakeaconnectionbelow).Nowwearereadytocalculatethegeometricproductofthemultivectors𝐴and𝐵.𝐴𝐵 = 𝛼� + 𝛼G𝒆G + 𝛼H𝒆H + 𝛼GH 𝒆G𝒆H 𝛽� + 𝛽G𝒆G + 𝛽H𝒆H + 𝛽GH 𝒆G𝒆H
=...multiplytermbytermandcombineliketerms...
= 𝛼�H + 𝛼GH + 𝛼HH − 𝛼GHH + 𝛼�𝛽G + 𝛼G𝛽� 𝒆G + 𝛼�𝛽H + 𝛼H𝛽� 𝒆H
+ 𝛼�𝛽GH + 𝛼GH𝛽� + 𝛼G𝛽H − 𝛼H𝛽G 𝒆G𝒆H .
Theresultisclearlya𝔾H-multivector(acombinationofscalar,vectorandbivector)whichproves
that𝔾Hisclosedalsoundergeometricproductandconfirmstheroleof𝔾Hastherightarenafor
geometricalgebra.
SIMULATINGCOMPLEXNUMBERSIN𝔾H
Let’snowconsidertheshockingfindingthatin𝔾Hwehavesomething,namelythebivector𝒆G𝒆H,
whichbehavesliketheimaginaryunit𝑖inthecomplexnumberalgebra.Bothsquareto−1.To
simplifythestudyofthisanalogyletusdenotethebivector𝒆G𝒆H = 𝒆G ∧ 𝒆Hbysymbol𝐼,thatis
𝐼 = 𝒆G𝒆H = 𝒆G ∧ 𝒆H.Thisbivectoriscalledtheunitpseudoscalarof𝔾H.(Unit,becauseitsnorm
𝐼 = 𝒆G 𝒆H = 1.)Sotheunitpseudoscalar𝐼isanelementof𝔾Hwhichsatisfies𝐼H = −1.
Forobviousreasonswearenowcurioustoinvestigate𝔾H-multivectorsoftheform𝛼 + 𝛽𝐼,where
𝛼and𝛽arescalarcoefficients.Considertwosuchobjects𝐴 = 𝛼G + 𝛼H𝐼and𝐵 = 𝛽G + 𝛽H𝐼and
addthem.Weget𝐴 + 𝐵 = (𝛼G + 𝛽G) + (𝛼H + 𝛽H)𝐼whichisamultivectorofthesameform.The
geometricproductof𝐴and𝐵is
𝐴𝐵 = (𝛼G + 𝛼H𝐼)(𝛽G + 𝛽H𝐼)
52
= 𝛼G𝛽G + 𝛼G𝛽H𝐼 + 𝛼H𝛽G𝐼 + 𝛼H𝛽H𝐼H [usethefact𝐼H = −1]
= (𝛼G𝛽G − 𝛼H𝛽H) + (𝛼G𝛽H + 𝛼H𝛽G)𝐼
beingagainamultivectorofthesameformas𝐴and𝐵.
Weconcludethatthe𝔾H-multivectors𝑍oftheform𝑍 = 𝛼 + 𝛽𝐼constitutea𝔾H-subdomain
whichisclosedunderallrelevantoperations(addition,scalarmultiplication,geometricproduct).
Thissubdomainclearlysimulatesℂ,thealgebraofcomplexnumbersthatwestudiedearlierin
thesenotes.Inthiswaythe2D-geometricproductcanreplacethecomplexproduct.Allconcepts
andresultsofcomplexalgebra(andanalysis)canbedevelopedwithintheframeworkofgeometric
algebrainasubdomainof𝔾H.Thisisaninterestingdemonstrationofthepowerandflexibilityof
theideasofgeometricalgebra.
ROTATIONINℝH
Weobservedthatthescalar-bivectorcombinations𝑍 = 𝛼 + 𝛽𝐼behavelikecomplexnumbers𝑧 =
𝑎 + 𝑏𝑖whenthecomplexproductisreplacedbythegeometricproduct.Yourememberthatthe
unitcomplexnumber𝑒o� = cos 𝜃 + 𝑖 sin 𝜃(callitrotor)wereusedtorotatealgebraicallyanyℝH-
vector𝒂 = 𝑎G𝒆G + 𝑎H𝒆Hbyanangle𝜃inthepositive(counterclockwise)direction.Thiswasdone
byexpressingthevector𝒂asacomplexnumber𝑎 = 𝑎G + 𝑎H𝑖andcalculatingthecomplex
product𝑒o�𝑎andfinallyturningtheresultbackintovectorform.
Letusseeifwecancopythistechniquewithournew𝔾H-rotorswhichareoftheanalogousform
𝑒§� = cos 𝜃 + 𝐼 sin 𝜃.Totestthisletustake𝜃 = 𝜋 4andapplytherotor𝑒§�totheℝH-vector
𝒂 = 𝒆G + 𝒆H.Tocalculatethecorrespondingproduct𝑒§�𝒂in𝔾Hwecompute,asthefirststepthe
products𝐼𝒆Gand𝐼𝒆H.Wehave
𝐼𝒆G = (𝒆G𝒆H)𝒆G = 𝒆G𝒆H𝒆G = −𝒆G𝒆G𝒆H = −1𝒆H = −𝒆H
and
𝐼𝒆H = (𝒆G𝒆H)𝒆H = 𝒆G𝒆H𝒆H = 𝒆G1 = 𝒆G.
Hence
𝑒§�𝒂 = cos 𝜃 + sin 𝜃 ∙ 𝐼 𝒆G + 𝒆H
= cos 𝜃 𝒆G + cos 𝜃𝒆H + sin 𝜃 𝐼𝒆G + 𝐼𝒆H
= cos 𝜃 𝒆G + cos 𝜃𝒆H + sin 𝜃 −𝒆H + 𝒆G
= cos 𝜃 + sin 𝜃 𝒆G + cos 𝜃 − sin 𝜃 𝒆H.
53
Nowfor𝜃 = 𝜋 4wehavecos 𝜃 = sin 𝜃 = 2 2andthereforetherotatedvectoris
𝑒§�𝒂 = 2𝒆G,
whichisthevectorobtainedbyrotatingouroriginalvector𝒂 = 𝒆G + 𝒆Hbytheangle−𝜋 4,that
is45°inthenegativedirection(clockwise).ItseemsthatmultiplyingtheℝH-vector𝒂bytherotor
𝑒§�fromtheleftrotatesthevectorbytheangle– 𝜃(i.e.by𝜃clockwise).Thisassumptioncanbe
easilyverifiedbyconsideringageneralℝH-vector𝒂 = 𝑟(cos 𝛼 𝒆G + sin 𝛼 𝒆H).Aroutine
calculationplussometrigonometryshowsthatthen𝑒§�𝒂 = 𝑟(cos(𝛼 − 𝜃) 𝒆G + sin(𝛼 − 𝜃) 𝒆H).
Itisnowfairlyobviousthattherotationofavector𝒂inthepositivedirection(counterclockwise)is
accomplishedbymultiplyingthevectorfromtheleftbytheoppositerotor𝑒�§�,sothatthe
rotatedvectorisexpressedby𝑒�§�𝒂.Again(exercise)youcanverifythisbycomputingthe
geometricproduct𝑒�§�𝒂forageneralℝH-vector𝒂 = 𝑟(cos 𝛼 𝒆G + sin 𝛼 𝒆H).Acareful
calculationshouldgivetheanswer𝑒�§�𝒂 = 𝑟(cos(𝛼 + 𝜃) 𝒆G + sin(𝛼 + 𝜃) 𝒆H).
Alternatively,youcandotherotationsbyrightmultiplications.Thenwehave𝒂𝑒§� = 𝑒�§�𝒂, [rotationof𝒂intopositivedirectionbytheangle𝜃],
𝒂𝑒�§� = 𝑒§�𝒂, [rotationof𝒂intonegativedirectionbytheangle𝜃].asyoucaneasilyseebyastraightforwardcomputation.
GEOMETRICALGEBRAIN𝔾M
Let’sconsiderthedomainℝMofvectorswiththeorthonormalbasis 𝒆G, 𝒆H, 𝒆M anditsmultivector
extension𝔾Mwiththebasis 1, 𝒆G, 𝒆H, 𝒆M, 𝒆M𝒆H, 𝒆G𝒆M, 𝒆H𝒆G, 𝒆G𝒆H𝒆M ofeightelements(oneforthe
scalarpart,threeforthevectorpart,threeforthebivector34partandoneforthetrivector35part).
Againasimplethoughtediouscomputationshowsthat𝔾Misclosedundergeometricproduct
(besidesbeingtriviallyclosedunderadditionandscalarmultiplication).So𝔾Misanappropriate
domainforthree-dimensional36geometricalgebra.
34Theorderoffactorsinthebasisbivectorsisnotimportantbecauseoftheorthogonalanti-commutativityofthegeometricproduct.Theorderchosenhereisconvenientinthenextchapter.35Anyorderoffactorsinthetrivector𝒆G𝒆H𝒆Mwouldbeequallygood(barthesign).36Thedomain𝔾Minitselfisthus8-dimensionalbutitisbuiltonthe3-dimensionaldomainℝMofvectors.
54
Abovewefoundthat𝔾Hcontainsaninterestingsubdomainofelements𝛼 + 𝛽𝒆G𝒆H = 𝛼 + 𝛽𝐼
(combinationsofscalarandbivector)whichbehavelikecomplexnumbers.Inthenextchapterwe
studythecorrespondingsubdomainof𝔾M.
SIMULATINGQUATERNIONSIN𝔾M
Consider𝔾M-scalar-bivector-combinations𝑃oftheform𝑃 = 𝛼� ∙ 1 + 𝛼G 𝒆M𝒆H + 𝛼H 𝒆G𝒆M + 𝛼M 𝒆H𝒆G = 𝛼� + 𝛼G𝐼 + 𝛼H𝐽 + 𝛼M𝐾wherewehaveadoptedasimplifiednotationforbivectors:𝒆M𝒆H = 𝐼,𝒆G𝒆M = 𝐽and𝒆H𝒆G = 𝐾.By
theproperty(6c)–orthogonalanticommutativity–itfollows(exactlyasin𝔾H)that𝐼H = 𝐽H =
𝐾H = −1whichbringstoourmindsthequaternionicimaginaryunits𝑖, 𝑗and𝑘.Alsowehave𝐼𝐽 = 𝒆M𝒆H 𝒆G𝒆M = 𝒆M𝒆H𝒆G𝒆M = 𝒆H𝒆G𝒆M𝒆M = 𝒆H𝒆G1 = 𝒆H𝒆G = 𝐾,
𝐽𝐼 = 𝒆G𝒆M 𝒆M𝒆H = 𝒆G𝒆M𝒆M𝒆H = 𝒆G1𝒆H = 𝒆G𝒆H = −𝒆H𝒆G = −𝐾.Similarcomputationsgivealso𝐽𝐾 = 𝐼and𝐾𝐽 = −𝐼,
𝐾𝐼 = 𝐽and𝐼𝐾 = −𝐽.
Theseresultsconfirmwhatweanticipated,namelythatthereisacloseanalogybetweenthe𝔾M-
bivectors𝐼, 𝐽, 𝐾andthequaternionicimaginaryunits𝑖, 𝑗, 𝑘.Sothemultivectorsoftheformof𝑃
areindeedanotherversionofquaternionsthatwestudiedearlier.Itiseasytoseethatthe
geometricproductoftwosuchmultivectors(scalar-bivectorcombinations)isagainofthesame
form.Thismeansthatthescalar-bivectorcombinationsconstituteaclosed4-Dsubdomain37of
𝔾Mandthissubdomainisessentiallyidenticalwiththedomainofquaternionsdiscussedearlier.
Thequaternionalgebracanthereforebeconsideredasaspecialcaseofthegeometricalgebrain
𝔾Minthesamewayasthecomplexalgebrawasaspecialcaseofthegeometricalgebrain𝔾H.
37Itisfour-dimensionaldomainbecauseitsmembers𝛼� + 𝛼G𝐼 + 𝛼H𝐽 + 𝛼M𝐾arespecifiedbyfourrealcoefficients𝛼J.
55
EXPRESSING𝒂 ∙ 𝒃AND𝒂 ∧ 𝒃INTERMSOFTHEGEOMETRICPRODUCT𝒂𝒃
Wehaveseenthatthecomplexandquaternionicproductsofvectorscanbeexpressedasspecial
casesofthegeometricproductin𝔾Hand𝔾M.Thisistrue,evenmoregenerally,forinner(scalaror
dot)andouter(wedge)products.
Infact,bytheverydefinitionofthegeometricproductwehave
𝒂𝒃 = 𝒂 ∙ 𝒃 + 𝒂 ∧ 𝒃
and
𝒃𝒂 = 𝒃 ∙ 𝒂 + 𝒃 ∧ 𝒂 = 𝒂 ∙ 𝒃 − 𝒂 ∧ 𝒃.[because𝒃 ∙ 𝒂 = 𝒂 ∙ 𝒃and𝒃 ∧ 𝒂 = −𝒂 ∧ 𝒃]
Hence𝒂𝒃 + 𝒃𝒂 = 2(𝒂 ∙ 𝒃)and𝒂𝒃 − 𝒃𝒂 = 2(𝒂 ∧ 𝒃)andconsequentlywehave
𝒂 ∙ 𝒃 =12 𝒂𝒃 + 𝒃𝒂
and
𝒂 ∧ 𝒃 =12 𝒂𝒃 − 𝒃𝒂 .
Weseethattheinnerproductandouterproductcanbeexpressedintermsofthegeometric
product.Thismightseemlikeacirculardefinitionbecausewedefinedthegeometricproductin
termsofinnerandouterproducts.However,therearecaseswheretheaboveexpressionsare
veryusefulbecausecalculationswiththegeometricproductareoftensimplerthanwiththeinner
orouterproducts.
Mostimportantly,theaboveexpressionsmakeitpossibletoturnthewholealgebraofvectorsand
multivectorsupsidedownbytakingthegeometricproductastheprimitiveconcept,astarting
pointfromwhicheverythingelseflowsout.Thisisthemodernapproachtogeometricalgebra:the
properties(1)–(8)listedabovearetakenasaxiomsforthegeometricproductwhichisthenused
todefineother,morespecialkindofproducts.
56
TRIVECTORSIN𝔾M
Thelastmemberinthe𝔾M-setofbasiselementsisthetrivector𝑇 = 𝒆G𝒆H𝒆Mwhichisalsocalled
thepseudoscalarin𝔾M.Ifyoutakeanythreevectors𝒂, 𝒃, 𝒄inℝMwhicharelinearlyindependent
(thatis,notinthesameplane)thentheirgeometricproduct𝒂𝒃𝒄isalsoatrivector.Aneasy
computationhowevershows38that𝒂𝒃𝒄 = 𝛼𝑇,where𝛼isascalar.Hencethereisessentiallyonly
onetrivector,namely𝑇,in𝔾M.Inthesetofeight𝔾M-basiselementsthereisonlyonescalar(1)
andonlyonetrivector(𝑇)whichisoneofthereasonswhy𝑇iscalledapseudoscalar.
MIRRORREFLECTIONOFANℝM-VECTORINAPLANE
ConsideranℝM-vector𝒂placedattheoriginandaplane𝑝passingthroughtheorigin.Theplaneis
specifiedbyitsunitnormalvector𝒏.Ourtasknowistofindanalgebraicexpressionsforthe
vector𝒂′whichisthereflectionof𝒂intheplane𝑝(seeFigure16.)
Figure16.The(red-framed)plane𝑝passesthroughtheorigin𝑂.Theunitvector𝒏isperpendiculartotheplane.Thevector𝒂ispositionedintheorigin𝑂andhascomponents𝒂𝒏and𝒂¯.Thecomponent𝒂𝒏isparallelto𝒏(thatis,alongtheunitnormalvector𝒏)andperpendiculartotheplane.Thecomponent𝒂¯isparalleltotheplane(thatis,alongtheplane)andperpendicularto𝒏.Wehave𝒂 = 𝒂𝒏 + 𝒂¯.Thevector𝒂� = −𝒂𝒏 + 𝒂¯isthereflectionof𝒂intheplane𝑝.
38Justwritethevectors𝒂, 𝒃, 𝒄intermsbasisvectors𝒆G, 𝒆H, 𝒆Mandmultiplytermbyterm(usingdistributivity).Allindividualproductsare(byorthogonalanticommutativity)equalto𝑇or– 𝑇,andcanthereforecollectedtogetherintoonesingleterm𝛼𝑇.
57
Wewillconstructageometric-algebraicexpressionforthereflectedvector𝒂�intermsofthe
originalvector𝒂andtheunitnormalvector𝒏whichspecifiesthe(mirror)plane𝑝.Asthefirst
stepwefindtheexpressionsfortheprojections𝒂𝒏and𝒂¯ofthevector𝒂onthenormalvector𝒏
andandtheplane𝑝.Westartourconstructionwiththeobservationthat𝒏H = 𝒏𝒏 = 1which
followsfromtheproperty(6c)andthefactthat𝒏isaunitvector,thatis 𝒏 = 1.Thereforewe
have
𝒂 = 𝒏H𝒂 = 𝒏𝒏𝒂 = 𝒏(𝒏𝒂)
= 𝒏(𝒏 ⋅ 𝒂 + 𝒏 ∧ 𝒂) [bydefinitionofgeometricproduct]
= 𝒏(𝒏 ⋅ 𝒂) + 𝒏(𝒏 ∧ 𝒂) [bydistributivityofgeometricproduct]
= 𝒂𝒏 + 𝒏(𝒏 ∧ 𝒂), [because𝒂𝒏 = 𝒏 𝒏 ⋅ 𝒂 = 𝒏( 𝒂 cos 𝜃)]
fromwhichitfollowsthattheplanarcomponent𝒂¯ofthevector𝒂hastheexpression
𝒂¯ = 𝒂 − 𝒂𝒏 = 𝒏(𝒏 ∧ 𝒂) .
Usingthefactthatthewedgeproduct𝒏 ∧ 𝒂canbeexpressedintermsofthegeometricproduct,
thatis𝒏 ∧ 𝒂 = GH(𝒏𝒂 − 𝒂𝒏),wecaneliminatethewedgeproductfromtheexpessionof𝒂¯and
writepurelyintermsofgeometricproduct
𝒂¯ = 𝒏 𝒏 ∧ 𝒂 = 𝒏12 𝒏𝒂 − 𝒂𝒏 =
12 𝒏𝒏𝒂 − 𝒏𝒂𝒏 =
12 𝒂 − 𝒏𝒂𝒏 .
Similarly,because𝒏 ∙ 𝒂 = G
H(𝒏𝒂 + 𝒂𝒏)wehaveforthenormalcomponent
𝒂𝒏 = 𝒏 𝒏 ⋅ 𝒂 =12𝒏 𝒏𝒂 + 𝒂𝒏 =
12 𝒏𝒏𝒂 + 𝒏𝒂𝒏 =
12 𝒂 + 𝒏𝒂𝒏 .
Nowwecanwritetheexpressionforthereflectedvector𝒂′asfollows
𝒂� = 𝒂¯ − 𝒂𝒏
=12 𝒂 − 𝒏𝒂𝒏 −
12 𝒂 + 𝒏𝒂𝒏
= −𝒏𝒂𝒏.
Wehaveprovedthefollowingresult.
58
”Sandwich”formulaforreflection
Ifavector𝒂isreflectedinamirrorplane𝑝whichisdefinedbyitsunitnormalvector𝒏,thenthereflectionvector𝒂′isgivenbythegeometricproduct 𝒂� = −𝒏𝒂𝒏.(Inwords:thereflectionof𝒂isobtainedby”sandwiching”𝒂between–𝒏and𝒏andconsideringthesandwichasageometricproduct.)
Theexpression–𝒏𝒂𝒏forthereflectedvectorseemsabitstrangeandsuspicious.Onemay
reasonablydoubtwhetheritisavectoratall!Letustestitinaconretecase.
Example.InℝMconsiderthevector𝒂 = 𝒆G + 𝒆H + 𝒆M.Findthereflectionof𝒂inthe𝑦𝑧-plane.
Solution:The𝑦𝑧-planeisdefinedbyitsunitnormalvector𝒏 = 𝒆G.Weusethesandwichformula
tocomputethereflectedvector𝒂′.
𝒂� = −𝒏𝒂𝒏 = −𝒆G(𝒆G + 𝒆H + 𝒆M)𝒆G
= (−𝒆G𝒆G − 𝒆G𝒆H − 𝒆G𝒆M)𝒆G [bracketmultipliedfromleftby𝒆G]
= (−1 − 𝒆G𝒆H − 𝒆G𝒆M)𝒆G [because𝒆G𝒆G = 1]
= −𝒆G − 𝒆G𝒆H𝒆G − 𝒆G𝒆M𝒆G [bracketmultipliedfromrightby𝒆G]
= −𝒆G + 𝒆G𝒆G𝒆H + 𝒆G𝒆G𝒆M [byorthogonalanticommutativity]
= −𝒆G + 1𝒆H + 1𝒆M [byassociativityandusing𝒆G𝒆G = 1]
= −𝒆G + 𝒆H + 𝒆M.
Voilá!Theanswerclearlyconformswithourgeometricintuition.(Justsitinthecornerofyour
roomandvisualizethesituation.)
Theaboveexamplehelpsustobelievethatthesandwichformulareallyproducesvectorsand
thesevectorsaretherequiredreflections.Indeed,itisfairlystraightforward(thoughabittedious)
exercisetoprove–followingthestepsoftheexample–thatforanyvectors𝒂and𝒃inℝM(or
generallyinℝJ)theexpression𝒃𝒂𝒃isalwaysavector.
59
WHYISTHESANDWICHFORMULABETTERTHANTHEOLDMETHODS?
Youmaywonderwhatadvantagewegainwiththesandwichformulacomparedwitholdmethods.
LookingattheFigure16aboveweseeeasilythat𝒂� = 𝒂 − 2𝒂𝒏 = 𝒂 − 2 𝒂 ∙ 𝒏 𝒏whichseemsa
simplerreflectionformula.Inthepreviousexamplewehad𝒂 = 𝒆G + 𝒆H + 𝒆Mand𝒏 = 𝒆G.Hence
𝒂 ∙ 𝒏 = 1andhence𝒂� = (𝒆G + 𝒆H + 𝒆M) − 2𝒆G = −𝒆G + 𝒆H + 𝒆M.Theoldmethodgavethesame
answerwithmuchlesswork!
However,theadvantageofthesandwichformulabecomesapparentifyouhavetocarryouta
chainofsuccessivereflections.Forexample,ifyoureflectthevector𝒂firstintheplane𝒏(i.e.the
planedefinedbytheunitnormalvector𝒏)andthenreflectthereflectedvector𝒃 = −𝒏𝒂𝒏in
anotherplane𝒎,thenthefinalresult𝒄willbe
𝒄 = −𝒎𝒃𝒎 = −𝒎 −𝒏𝒂𝒏 𝒎 = 𝒎𝒏𝒂𝒏𝒎
andifyoudoonemorerefletionintheplanedefinedbytheunitnormalvector𝒑youwillendup
tothevector
𝒅 = −𝒑𝒎𝒏𝒂𝒏𝒎𝒑
andsoon.Thesandwichformulaisaneffective(andcomputer-friendly)machineforcarryingout
successivereflections.Noticethatthefrontsidelayer𝒑𝒎𝒏andthebacksidelayer𝒏𝒎𝒑ofthe
chainedsandwichareinreversedorder.
ROTATIONOFAVECTORINℝM
Inchapter7westudiedtherotationofanℝM-vectoraroundagivenaxisandfoundaquaternionic
formulaforcomputingtherotatedvector.Nowweapproachthesameproblemwithhelpofthe
geometricproduct.Thisapproachisrelatedtoso-calledCartan-DieudonnéTheoremprovedby
thesetwoFrenchmathematiciansin1937.AsimplifiedversionofthistheoremsaysthatinℝM
everyrotationaroundtheorigincanbedonebysuccessivereflectionsinplanes.Belowwewill
presentasimplegeometricargumenttosupportthisstatement.
Considertwoorigin-basedunitvectors𝒎and𝒏withanintermediateangle𝜔.Thesevectors
defineaplane(throughtheorigin)inℝM.(Wecanalsosaythattheouterproduct𝒎∧ 𝒏defines
thisplane.)Nowthesevectors(ortheirouterproduct𝒎∧ 𝒏)canbeusedasaspecificationofa
60
rotationofanyorigin-basedvector𝒂aroundthenormaloftheplane.Wewillnowseethatsucha
rotationcanbeviewed(inthespiritofCartan-DieudonnéTheorem)asaresultoftwosuccessive
reflectionsinplanesperpendiculartothevectors𝒎and𝒏(seeFigure17).
Figure17.Thebasicplane𝒎∧ 𝒏throughtheorigin𝑂isdefinedbytheunitvectors𝒎and𝒏withanintermediateangle𝜔.Themirrorplanes𝑝¡and𝑝Jareperpendicularto𝒎and𝒏andintersectthebasicplanealongtheblueandbrowndashedlineswhichmakethesameangle𝜔witheachother.(Oftheseplanesonlythedashedintersectionlinesareshown.)Theprocessstartswithanarbitraryvector𝒂.Firstthevector𝒃isobtainedbyreflecting𝒂intheplane𝑝¡.Thenthevector𝒄isobtainedbyreflecting𝒃intheplane𝑝J.Thevector𝒂anditsreflections𝒃and𝒄haveobviouslyequallengthsandthemakeequalangleswiththebasicplane.Hencetheirtippointsareatequaldistancesfromthebasicplaneandthecircle(ellipticinperspective)throughthesetippointsislocatedinaplaneparalleltothebasicplane.Inthesituationshowninthefigure17wehave(bythesandwichformula)
𝒄 = 𝒏𝒃𝒏 = 𝒏𝒎𝒂𝒎𝒏 = 𝑅𝒂𝑅´,
wherewehaveusedthenotations𝒏𝒎 = 𝑅and𝒎𝒏 = 𝑅´.Theformeriscalledrotorbecause,as
wewillshortlysee,itiscloselyrelatedtorotation.Bydefinition,therotor𝑅 = 𝒏𝒎isageometric
productofunitvectorswhichdefinethemirrorplanes.Noticethattheorderofvectorsinthe
61
rotorproduct𝒏𝒎isoppositetotheorderofreflections:firstby𝒎andthenby𝒏.Thelatterrotor
𝑅´(wheretheupperindexispronounced”dagger”)is–forobviousreasons–calledthereverseof
𝑅.Noticethat 𝑅´ ´ = 𝑅.
FromFigure17itisclearthatthevector𝒄thatresultsfrom𝒂bydouble-reflectioncanalsobe
obtainedbyrotatingthevector𝒂aroundtheaxispassingthroughtheorigin𝑂andthe
centerpointofthecircleabove(alineperpendiculartothebasicplane𝒎∧ 𝒏).IntheFigure18
belowweshowthattherotationanglefrom𝒂to𝒄isequalto2𝜔,where𝜔istheanglebetween
𝒎and𝒏.
Figure18.TheplaneoftheuppercircleinFigure17viewedfromabove.Thedashedlinesshowthelinesofintersectionwiththeverticalmirrorplanes𝑝¡and𝑝J.Theanglebetweentheselinesisequalto𝜔,theanglebetweenthevectors𝒎and𝒏inFigure17.Points𝐴, 𝐵, 𝐶arethetippointsofthevectors𝒂, 𝒃, 𝒄.Thelines𝑝¡and𝑝Jbisecttheangles𝐴𝑂𝐵 = 2𝛽and𝐵𝑂𝐶 = 2(𝛽 + 𝜔).Fromthecentraltrianglesofthecircleweseethatthesmallcentralanglebetweentheline𝑝Jandtheradius𝑂𝐴isequalto𝜔 − 𝛽.Thecentralangle𝐴𝐶𝑂isthenequalto 𝛽 + 𝜔 + 𝜔 − 𝛽 = 2𝜔.Thisistheangleofrotationfrom𝐴to𝐶.
Weconcludethattheformula𝒂′ = 𝑅𝒂𝑅´rotatesthevector𝒂tovector𝒂′.Theangleofrotation
istwicetheanglebetweentheinitialunitvectors𝒎and𝒏fromwhichtherotor𝑅 = 𝒏𝒎was
built.Alsonoticethattheorientationoftherotationangleisfrom𝒎to𝒏.If,forinstance,we
wanttoconstructarotorwhichrotatesvectorsby+120°withrespectofagivenplanethenwe
62
mustconstructourrotorasthegeometricproductofunitvectors𝒎and𝒏inthisplanewithan
intermediateangle+60°(countedfrom𝒎to𝒏).
Note:Thetraditionalmethods39rotatevectorsaroundagivenaxis.Ingeometricalgebrarotationisdefinedwithrespecttoa(basic)planewhichisspecifiedbyanorientedpairofunitvectors𝒎and𝒏,i.e.byabivector𝒎∧ 𝒏.Thedifferencebetweentheseapproachesis,tosomeextent,amatteroftaste,sincetherotationaxisandrotationplanearecloselyconnected(beingperpendicularagainsteachotherinℝM).Themainadvantagesoftherotortechniqueingeometricalgebraarethat(1)itoffersaneatlogicforcomputerimplementationand(2)itgeneralizesdirectlytoalldimensionshigherthanthree.
Technicalremark:Inthefollowingsectionweneedtwopropertiesofthescalarproductandtheouterproductofourunitvectors𝒎and𝒏withtheintermediateangle𝜔.Firstwecomputeeasilythescalarproduct𝒏 ∙ 𝒎 = 𝒏 ∙ 𝒎 ∙ cos𝜔 = 1 ∙ 1 ∙ cos𝜔 = cos𝜔.Secondlywecomputethe(geometric)squareoftheouterproduct 𝒎∧ 𝒏 H,thatisthegeometricproductofthebivector𝒎∧ 𝒏withitself.We’llusefivefacts:(1)𝒎∧ 𝒏 = − 𝒏 ∧𝒎 ,i.e.theanticommutativityoftheouterproduct,(2)𝒎∧ 𝒏 = 𝒎𝒏−𝒎 ∙ 𝒏,(3)𝒏 ∧𝒎 = 𝒏𝒎− 𝒏 ∙ 𝒎 =𝒏𝒎−𝒎 ∙ 𝒏,(4)𝒏𝒎𝒎𝒏 = 𝒏 𝒎𝒎 𝒏 = 𝒏1𝒏 = 𝒏𝒏 = 1and(5)𝒏𝒎+𝒎𝒏 = 2(𝒎 ∙ 𝒏)Facts(2),(3)and(5)followfromthedefinitionofgeometricproductandthecommutativityofscalarproduct.Fact(4)followsfromtheproperty(6c)ofgeometricproduct.Nowwehave(𝒎 ∧ 𝒏)H = 𝒎 ∧ 𝒏 𝒎 ∧ 𝒏 = − 𝒏 ∧𝒎 𝒎 ∧ 𝒏 [byfact(1)]= − 𝒏𝒎−𝒎 ∙ 𝒏 𝒎𝒏 −𝒎 ∙ 𝒏 [byfacts(2)and(3)]= −[𝒏𝒎𝒎𝒏− 𝒎 ∙ 𝒏 𝒏𝒎− 𝒎 ∙ 𝒏 𝒎𝒏 + 𝒎 ∙ 𝒏 𝟐] [bydistributivityofg.p.]= −𝒏𝒎𝒎𝒏+ 𝒎 ∙ 𝒏 𝒏𝒎+𝒎𝒏 − 𝒎 ∙ 𝒏 𝟐] [byregroupingterms]= −1 + 2 𝒎 ∙ 𝒏 𝟐 − 𝒎 ∙ 𝒏 𝟐] [byfacts(4)and(5)]= −1 + 𝒎 ∙ 𝒏 𝟐] = −1 + cosH 𝜔= − sinH 𝜔. [bytrigonometry]Henceweconcludethattheunitvectors𝒏and𝒎havethefollowingproperties:(P1)𝒏 ∙ 𝒎 = cos𝜔and(P2)(𝒏 ∧𝒎)H = −sinH 𝜔 = (𝒎 ∧ 𝒏)Hwhere𝜔istheanglebetween𝒏and𝒎.Notice,asaspecialcase,thatifthevectors𝒏and𝒎areorthogonal(𝜔 = 𝜋/2)then𝒏 ∙ 𝒎 = 0and(𝒏 ∧𝒎)H = −1.
39Themethodofso-calledEulerangles–inventedbytheSwissmathematicianLeonhardEuler(1707-1783)–hasbeenthemostinfluential.
63
EXPONENTIALFORMOFTHEROTOR𝑅
Letagain𝒎and𝒏beunitvectorsasabovewiththeintermediateangle𝜔(from𝒎to𝒏whichis
takenasthepositiveorientation).Thentherotor𝑅 = 𝒏𝒎definedbythesevectorscanbe
developedasfollows:𝑅 = 𝒏𝒎 = 𝒏 ∙ 𝒎 + 𝒏 ∧𝒎 [bydefinitionofgeometricproduct]
= cos𝜔 + 𝒏 ∧𝒎 [by(P1)above]
= cos𝜔 −𝒎 ∧ 𝒏 [byanticommutativityof∧]
= cos𝜔 − G¶·¸¹
(𝒎 ∧ 𝒏) sin𝜔. [ G¶·¸¹
∙ sin𝜔 = 1]Nowdenote G
¶·¸¹𝒎 ∧ 𝒏 = 𝐵,whichisabivector.Weobservethatbytheproperty(P2)
establishedabovethesquareofthisbivectoris
𝐵H =1
sin𝜔
H
(𝒎 ∧ 𝒏)𝟐 =1
sinH 𝜔 ∙ − sinH 𝜔 = −1,
andhence𝐵isabivectorwiththeproperty𝐵H = −1whichputs𝐵inthesamecategorywiththe
imaginaryunit𝑖ofthecomplexplaneℂaswellasthegeneralimaginaryunitquaternions𝑢
discussedaboveinchapters5and7.Nowtheexpressionoftherotorsimplifiesto𝑅 = cos𝜔 − 𝐵 sin𝜔.Butthisexpressionhasthesameformthatwehaveseenbeforeinthecontextofcomplex
numbersorquaternions,namelytheformcos𝜔 − 𝑖 sin𝜔which,aswelearnedinchapter5,can
bewrittenintheexponentialformcos𝜔 − 𝑖 sin𝜔 = 𝑒�o¹.Thisresult40wascalledEuler’sformula
anditfollowedfromthefactthat𝑖H = −1whichisequallytruewhenwereplace𝑖by𝐵.
Thereforewecanwritetherotor𝑅intheexponentialform𝑅 = cos𝜔 − 𝐵 sin𝜔 = 𝑒�º¹.Itisaneasyexercisetoshowthat𝑅´ = 𝒎𝒏,thereverseof𝑅 = 𝒏𝒎,canbewrittenanalogously𝑅´ = cos𝜔 + 𝐵 sin𝜔 = 𝑒º¹.
40ItwascalledEuler’sequation.
64
Therotationformula𝒂� = 𝑅𝒂𝑅´cannowbewrittenintheform𝒂� = 𝑒�º¹𝒂𝑒º¹.Thisformula
hastheadvantagethattherotationangle2𝜔isexplicitlyvisible.(Rememberthattherotation
angleistwicetheanglebetween𝒎and𝒏.)Wecannowrewritetherotationformulaasfollows.
Geometricsandwichformulaforrotatingvectorswithrespecttoabasicplane
Letabasicplanethroughtheoriginbedefinedbytwounitvectors𝒎and𝒏withanintermediate
angle𝜃/2.Let𝐵 = 𝒎 ∧ 𝒏 / sin 𝜃/2 betherespectiveunitbivectordefinedby𝒎and𝒏.Let
𝑅 = 𝒏𝒎 = 𝑒�º �/H and𝑅´ = 𝒎𝒏 = 𝑒�º �/H .Ifanarbitraryvector𝒂isrotatedbytheangle𝜃intheplaneparallel41totothebasicplanethen𝒂
transformsintotherotatedvector𝒂′whichisgivenbytheformula𝒂� = 𝑅𝒂𝑅´ = 𝑒�º �/H 𝒂𝑒º �/H ,wherethemultiplicationsaregeometric.
NB.Itisimportanttounderstandthatintheexponentialforms𝑅 = 𝑒�º �/H and𝑅´ = 𝑒º �/H oftherotoranditsreversetheunitbivector𝐵isnotdirectlyconnectedtothevectors𝒎and𝒏sinceitcanbeanyunitbivectordefiningthesameplanethat𝒎and𝒏define.Therotationangleiscodedexplicitlyintheparameter𝜃(seeexample2below).
Wenoticethattheexponentialrotationformulaaboveisverysimilartothequaternionicone𝑞� = 𝑒�(� H) ∙ 𝑞 ∙ 𝑒� �� H ,whichwepresented(withoutanyjustification)inchapter5.Themaindifferenceisthatthelatter
oneisonlyvalidinℝMwhiletheformeristrueinhigherdimensionsaswell.
Example1:Letusfixthebasicplaneofrotationtobeforsimplicitythe𝑥𝑦-planeandchoosein
thisplanetwounitvectors𝒎and𝒏withanintermediateangle𝜃 2 = 45°tospecifytherotation
bytheangleof90°.Set,forexample,𝒎 = 𝒆Gand𝒏 = GH(𝒆G + 𝒆H).Fromthemweconstructthe
rotor𝑅anditsreverse𝑅´:
41Thatis,thetailof𝒂sitsputintheoriginwhileitstiprotatesbytheangle𝜃aroundthenormallineofthebasicplane.
65
𝑅 = 𝒏𝒎 =12𝒆G + 𝒆H 𝒆G =
12𝒆G𝒆G + 𝒆H𝒆G =
121 + 𝒆H𝒆G =
121 − 𝒆G𝒆H ,
𝑅´ = 𝒎𝒏 =12𝒆G 𝒆G + 𝒆H =
12𝒆G𝒆G + 𝒆G𝒆H =
121 + 𝒆G𝒆H .
Astheinitialvectorwechoose𝒂 = 𝒆G + 𝒆Mwhichwewanttorotateby90°inthedirection𝒎to
𝒏ontheplanedeterminedbythem.Therotatedvector𝒂′is𝒂′ = 𝑅𝒂𝑅´ = 𝑅(𝒆G + 𝒆M)𝑅´
=121 − 𝒆G𝒆H (𝒆G + 𝒆M)
121 + 𝒆G𝒆H
=12 1 − 𝒆G𝒆H 𝒆G + 𝒆M 1 + 𝒆G𝒆H [dothefirstproduct]
=12 (𝒆G + 𝒆M − 𝒆G𝒆H𝒆G − 𝒆G𝒆H𝒆M) 1 + 𝒆G𝒆H [permute, useanticomm. ]
=12(𝒆G + 𝒆M + 𝒆G𝒆G𝒆H − 𝒆G𝒆H𝒆M) 1 + 𝒆G𝒆H [simplify:𝒆G𝒆G = 1]
=12(𝒆G + 𝒆H + 𝒆M − 𝒆G𝒆H𝒆M) 1 + 𝒆G𝒆H [dotheproduct]
=12 𝒆G + 𝒆H + 𝒆M − 𝒆G𝒆H𝒆M + 𝒆G𝒆G𝒆H + 𝒆H𝒆G𝒆H + 𝒆M𝒆G𝒆H − 𝒆G𝒆H𝒆M𝒆G𝒆H [simplify]
=12 𝒆G + 𝒆H + 𝒆M − 𝒆G𝒆H𝒆M + 𝒆H − 𝒆G + 𝒆G𝒆H𝒆M + 𝒆M [simplify]
=12 2𝒆H + 2𝒆M
= 𝒆H + 𝒆M,which,indeed,isthecorrectanswerasyoucaneasilyvisualize(sittinginacorneragain).
Remark1:Wehavefreedomtochoosethepairofinitialvectors𝒎and𝒏aslongastheyareunitvectors42andtheanglebetweenthem(from𝒎to𝒏)is45°.So,inthepreviousexamplewecould
havechosen,say,𝒎 = GH(𝒆G + 𝒆H)and𝒏 = 𝒆Hwithoutaffectingthefinalanswer𝒂′.
Remark2:Fromtheexampleweseethatthecomputationscanbe(eveninthisdeliberatelytailoredsimplecase)quitetedious.However,theprocessitselfisperfectlymechanisticandcanbeeasilyprogrammedonacomputerwhichproducestheanswerinafractionofasecond.
42Infact,wecouldchooseanyvectors𝒎and𝒏suchthat 𝒎 𝒏 = 1andtheangleis45°.
66
Example2:Letusdothesamerotationasinexample1abovebutusingnowtheexponential
versionsoftherotor𝑅anditsreverse.Fortheroleof𝐵wechoosethesimplestpossibleunit
bivectorwhichrepresentsthebasicplane(herethe𝑥𝑦-plane).Solet𝐵 = 𝒆G𝒆H = 𝒆G ∧ 𝒆Hbeour
choice.This𝐵isaunitbivectorbecause 𝐵 = 𝒆G ∧ 𝒆H = 𝒆G ∙ 𝒆H ∙ sin 90° = 1andithasalso
therequiredimaginaryunitproperty43since𝐵H = 𝒆G𝒆H𝒆G𝒆H = −𝒆G𝒆H𝒆H𝒆G = −1.So𝐵isfinefor
theexponentialconstructionoftherotor𝑅anditsreverse𝑅´.Fortherotationangle𝜃 = 45°we
have:
𝑅 = 𝑒�º �/H = cos𝜃2 − 𝐵 sin
𝜃2 =
12−12𝒆G𝒆H =
12(1 − 𝒆G𝒆H)
and
𝑅´ = 𝑒º �/H = cos𝜃2 + 𝐵 sin
𝜃2 =
121 + 𝒆G𝒆H .
Consequentlythegivenvector𝒂 = 𝒆G + 𝒆Mrotatesto𝒂� = 𝑅𝒂𝑅´ = 𝑒�º �/H 𝒂𝑒º �/H
=121 − 𝒆G𝒆H 𝒆G + 𝒆M
121 + 𝒆G𝒆H [dothefirstproduct]
=12 𝒆G + 𝒆M − 𝒆G𝒆H𝒆G − 𝒆G𝒆H𝒆M 1 + 𝒆G𝒆H [𝒆G𝒆H𝒆G = −𝒆G𝒆G𝒆H = −𝒆H]
=12 𝒆G + 𝒆M + 𝒆H − 𝒆G𝒆H𝒆M 1 + 𝒆G𝒆H [dothesecondproduct]
=12 𝒆G + 𝒆M + 𝒆H − 𝒆G𝒆H𝒆M + 𝒆G𝒆G𝒆H + 𝒆M𝒆G𝒆H + 𝒆H𝒆G𝒆H − 𝒆G𝒆H𝒆M𝒆G𝒆H
=12 𝒆G + 𝒆M + 𝒆H − 𝒆G𝒆H𝒆M + 𝒆H + 𝒆G𝒆H𝒆M − 𝒆G + 𝒆M
=12 2𝒆H + 2𝒆M
= 𝒆H + 𝒆M,whichcoincideswiththeanswerofexample1.
Remark.Theadvantageoftheexponentialversionoftherotoristhattherolesofthebasicplane𝐵(rotationplane)andtherotationangle𝜃havebeenseparatedandwecanfreelyrepresenttheplanebyanyappropriate44bivector𝐵.43ThispropertyisnecessarytomaketheEulerequation(𝑒ºÁ = cos 𝛼 + 𝐵 sin 𝛼)true.44Infact,wecouldconstructavalidbivector𝐵fromanynon-parallelvectors𝒂and𝒃inthebasicplanebysetting𝐵 = (𝒂 ∧ 𝒃)/( 𝒂 𝒃 sin 𝛾)where𝛾istheanglebetween𝒂and𝒃.
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10.SCALARSANDPSEUDOSCALARS
Werememberthatall𝔾Hmultivectorsareoftheform𝑀 = 𝑎� + 𝑎G𝒆G + 𝑎H𝒆H + 𝑎GH𝒆G𝒆Hwhere
𝑎�arescalarcoefficientsandthevectors𝒆�areorthonormalbasisvectorsforℝH,themother
vectorspaceintwodimensions.Weseethatthemultivectorspace𝔾Hisfour-dimensional45
becauseallelementsof𝔾Hare(linear)combinationsfourbasicelements 1, 𝒆G, 𝒆H, 𝒆G𝒆H .These
elementsrepresentdifferentgradesofmultivectors.Thegrade0element1generatesallscalars
(i.e.realnumbers),thegrade1elements𝒆Gand𝒆Hgenerateallvectors(allofℝH)andthegrade2
element𝒆G𝒆Hgeneratesallbivectors.Wefoundearlierthatthesefourbasiselementsare
sufficientforconstructingeverythingin𝔾H.Thisissobecauseallpossibletrivectors(like𝒆G𝒆H𝒆G)
mustnecessarilyhavetwoidenticalfactors(here𝒆G)andwillthereforereducetovectors(likein
ourexample𝒆G𝒆H𝒆G = −𝒆G𝒆G𝒆H == −𝒆H).Thesameisgenerallytrueforall𝑛-vectors(for𝑛 ≥ 3)
in𝔾H.Bytherulesofgeometricproducttheyreduceintoscalarsorvectorsorbivectors.
Wenoticeanice1-2-1-symmetryintheset 1, 𝒆G, 𝒆H, 𝒆G𝒆H of𝔾H-basiselements.Thereisone
elementofgrade0,twoelementsofgrade1andoneelementofgrade2.Thebasiselements1
and𝒆G𝒆Hareinsymmetricpositions.Indeed,theyhavemoreincommon?Botharesovereign
rulersoftheirsubdomains:grade0elementsandgrade2elements.Allscalars𝑥in𝔾Harescalar
multiplesof1(𝑥 = 𝑥 ∙ 1)andallbivectors𝒂 ∧ 𝒃in𝔾Harescalarmultiplesof𝒆G𝒆H = 𝒆G ∧ 𝒆H.To
seethatthelatteristruelet𝒂 = 𝑎G𝒆G + 𝑎H𝒆Hand𝒃 = 𝑏G𝒆G + 𝑏H𝒆H.Then
𝒂 ∧ 𝒃 = (𝑎G𝒆G + 𝑎H𝒆H) ∧ (𝑏G𝒆G + 𝑏H𝒆H)
= 𝑎G𝑏G𝒆G ∧ 𝒆G + 𝑎G𝑏H𝒆G ∧ 𝒆H + 𝑎H𝑏G𝒆H ∧ 𝒆G + 𝑎H𝑏H𝒆H ∧ 𝒆H
= 𝑎G𝑏H𝒆G ∧ 𝒆H + 𝑎H𝑏G𝒆H ∧ 𝒆G[𝒆G ∧ 𝒆G = 𝒆H ∧ 𝒆H = 0]
= 𝑎G𝑏H𝒆G ∧ 𝒆H − 𝑎H𝑏G𝒆G ∧ 𝒆H[𝒆H ∧ 𝒆G = −𝒆G ∧ 𝒆H]
= 𝑎G𝑏H − 𝑎H𝑏G 𝒆G ∧ 𝒆H
= 𝑎G𝑏H − 𝑎H𝑏G 𝒆G𝒆H.
45Thisiswhythenotation𝔾Hmayseemmisleadingas𝔾Hisnottwo-dimensional.Ingeneral𝔾Jhas2Jdimensions.Theindex2in𝔾HonlyindicatesthatthemothervectorspaceisℝH.(Toavoidconfusionsomeauthorshavechosendifferentnotationslike𝔾Hforthemultivectorspace.)
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Soweseethateverybivector𝒂 ∧ 𝒃isascalarmultipleofthebasicbivector𝒆G𝒆H = 𝒆G ∧ 𝒆H.We
observeasimilaritybetweenscalarsandbivectorsin𝔾H.Everyscalarisobtainedfromtheunit
scalar1byscalarmultiplicationandeverybivectorisobtainedfromtheunitbivector𝒆G𝒆H,which
isthereforecalledaunitpseudoscalar,oftendenotedby𝐼.Noticethatthisisnottrueforthe
vectors(grade1objects)in𝔾H.Thereisnosinglevectorwhichgeneratesallothervectorsasscalar
multiples.Whilethevectorsforma2-dimensionalsubspaceof𝔾H,thescalarsandpseudoscalars
form1-dimensionalsubspacesof𝔾H.
Thenotionofpseudoscalargeneralizestohigherdimensionsaswell.In𝔾M,forexample,thesetof
eightbasiselementsis 1, 𝒆G, 𝒆H, 𝒆M, 𝒆G𝒆H, 𝒆H𝒆M, 𝒆M𝒆G, 𝒆G𝒆G𝒆H ofwhichthefirst,1,istheunit
scalarandthelastistheunitpseudoscalar𝐼 = 𝒆G𝒆H𝒆M.Againitcanbeshownbyastraightforward
computationthateverytrivector𝒂 ∧ 𝒃 ∧ 𝒄in𝔾Misascalarmultipleoftheunitpseudoscalar𝐼.
Anotherinterestingpropertyofthe𝔾Junitpseudoscalaristhatitssquareisascalar,more
precisely𝐼H = ±1,dependingon𝑛.Forexample,forthe𝔾Hunitpseudoscalar𝐼 = 𝒆G𝒆Hwehave
𝐼H = 𝒆G𝒆H𝒆G𝒆H = −𝒆G𝒆H𝒆H𝒆G = −𝒆G𝒆G = −1.Thesameistrueforthe𝔾Munitpseudoscalar𝐼 =
𝒆G𝒆H𝒆Masyoucaneasilycheck.Agoodexerciseistocomputethesquareofthe𝔾Yunit
pseudoscalar𝐼 = 𝒆G𝒆H𝒆M𝒆Y.
Ifyougodeeperintothetheoryandapplicationsofgeometricalgebrayou’llfindanumberof
interestingpropertiesandusesofpseudoscalars.
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11.FINALREMARKS
Wehavenowcompletedashortexcursiontogeometricalgebra.ItwasWilliamKingdonClifford
(1845-1879)whoinitiatedthisfieldbydefiningthenotionsofgeometricproductandmultivectors.
HisworkwasbasedonthestudiesofWilliamRowanHamilton(1805-1865)–thefatherof
quaternions–andHermannGüntherGrassmann(1809-1877)whointroducedtheconceptof
outerproduct.BeforethemCarlFriedrichGauss(1777-1855)hadalreadydefinedthebasicideas
ofvectoralgebraandvectorcalculus(thelatterreferstotheconceptsofdifferentiationand
integrationasappliedtovectorfunctions).Laterinthe19thcenturyJosiahWillardGibbs(1839-
1903)adaptedandrefinedtheconceptsvectoralgebraforthepurposesofphysics.Inhis
approachthescalar(dot)andvector(cross)productsplayedtheminorrolewhileGrassmann’s
andClifford’sextensionstoouterandgeometricproductsturnedoutunnecessarilycomplicated
andtediousforhandcalculations.Gibb’sformulationofvectoralgebrabecamesoonverypopular
amongscientistsand(themoreabstract)innovationsofGrassmannandCliffordfellintooblivion.
However,withtheadventofcomputersintheendof20thcentury,theywerefoundagainand
developedfurtheraseffectivetoolsincomputergraphics,spacenavigationandphysicsingeneral.
Thenotionsofouterandgeometricproductofferednewconceptualclarityandflexibilityin
studyinggeometrictransformations(likereflectionsandrotations)andphysicalsystems.With
superfastcomputersthetediouscalculationswerenomoreaproblem.
Theseintroductorynoteshavebroughtyoutotheentranceroomofgeometricalgebra.Thebasic
conceptshavebeendefinedandsomeoftheirapplicationshavebeenconsidered,particularlyin
dimensions2and3.Muchmore,especiallyforhigherdimensions,hasbeendevelopedduringthe
lasttwoorthreedecades.Whenpreparingthesenotestheauthorhasbeendigginginformation
fromWikipediaandotherinternetsources(searchwords”geometricalgebra”).Themostuseful,
however,havebeenthetwobookslistedbelow.
C.Doran&A.Lasenby:GeometricAlgebraforPhysicists,CambridgeUniversityPress,2013(6thprinting),578pages.
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Thisbookpresentsthetheoryofgeometricalgebraandanalysisandsurveysextensivelyitsphysicalapplicationsinclassicalmechanics,specialandgeneralrelativity,electromagnetismandquantumtheory.Someapplicationsrequirefairlyadvancedbackgroundknowledgeoftherespectiveareas.
A.Macdonald:LinearandGeometricAlgebra,AlanMacdonald,2010(3rdprinting),208pages.Thisbookisasystematicmathematicalintroductiontolinearalgebraandgeometricalgebra.Thetreatmentisconcisebutaccessible.