Whatisgeometricalgebra?

PART1

Teuvo Laurinolli

2016

teuvo.laurinolli@gmail.com

2

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

3

0.ALKUSANAT/FOREWORDTämätekstionkoostelukuvuonna2015-2016esittämistäniviikkotarinoistaOulunLyseonlukion

matematiikankerhossa(lempinimeltäänGalois-kerho1).Esitintarinatsuomeksi,muttatekstion

englanninkielinenajatellenmahdollisiatuleviakinkerholaisia,joidensuomenkielentaitoon

puutteellinen.

LisäämälläotsikkoonmerkinnänPart1(Osa1)varaudunmahdollisuuteenjatkaageometrisen

algebrankäsittelyäpidemmällekin.

Tekstiiniliittyvätkommentitovattervetulleitakansisivullailmoitettuunsähköpostiosoitteeseen.

/////

ThesenotessummarizetheseriesoftalksIgaveinthemathclub(nicknamedGalois1WEWclub)of

OulunLyseo(aseniorhighschoolinOulu,Finland).

Theword”Part1”attheendofthetitleindicatesthatImightdecidetogoondeeperintothe

secretsofgeometricalgebra.

Iwishallremarksandcommentswelcometomyemail-addressgivenonthetitlepage.

Oulu,Finland

January2016

TeuvoLaurinolli

1ÉvaristeGalois(1811-1832),Frenchmathematicianwhosediscoveriesopenedthewaytoacomprehensivetheory–nowknownasGaloistheory–ofpolynomialequations.Hewasbadlywoundedinaduellanddiedattheageof20.

4

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.

5

proceedtothestandardvectoralgebraintwoandthreedimensions(planeandspace)aswellas

higherdimensions.Whiletheoperationofadditionhasanaturalandsimpledefinitionforvectors

thereareanumberofdifferentoptionsfortheoperationofmultiplication(product)ofvectors.

Theseincludescalar,complex,vector,quaternionic,outerandgeometricproducts.Wewill

considereachoftheseoptions–theirtechnicalpropertiesaswellasadvantagesand

disadvantages.OurfinalunifyingextensionisthegeometricalgebraalreadyinventedbyClifford

butforgottenforacenturyuntilrecentlywhenitwasfoundausefultoolincomputergame

programmingandphysics.

6

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.

7

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.

8

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.

9

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.

10

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).

11

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.

12

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.

13

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

14

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.

15

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.

44

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).

45

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).

49

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.

67

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.)

68

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.

69

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.

70

Thisbookpresentsthetheoryofgeometricalgebraandanalysisandsurveysextensivelyitsphysicalapplicationsinclassicalmechanics,specialandgeneralrelativity,electromagnetismandquantumtheory.Someapplicationsrequirefairlyadvancedbackgroundknowledgeoftherespectiveareas.

A.Macdonald:LinearandGeometricAlgebra,AlanMacdonald,2010(3rdprinting),208pages.Thisbookisasystematicmathematicalintroductiontolinearalgebraandgeometricalgebra.Thetreatmentisconcisebutaccessible.