The scalar product mcTYscalarprod Oneofthewaysinwhichtwovectorscanbecombinedisknownas the scalarproduct
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The scalar product mcTYscalarprod Oneofthewaysinwhichtwovectorscanbecombinedisknownas the scalarproduct

When wecalculatethescalarproductoftwovectorstheresultas thenamesuggestsisascalarrather thanavector Inthisunityouwilllearnhowtocalculatethescalarproduc tandmeetsomegeometricalappli cations Inordertomasterthetechniquesexplainedhereitisvitalt hatyouunde

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The scalar product mcTYscalarprod Oneofthewaysinwhichtwovectorscanbecombinedisknownas the scalarproduct




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The scalar product mc-TY-scalarprod-2009-1 Oneofthewaysinwhichtwovectorscanbecombinedisknownas the scalarproduct .When wecalculatethescalarproductoftwovectorstheresult,as thenamesuggestsisascalar,rather thanavector. Inthisunityouwilllearnhowtocalculatethescalarproduc tandmeetsomegeometricalappli- cations. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: definethescalarproductoftwovectors

statesomeimportantpropertiesofthescalarproduct calculatethescalarproductwhenthetwovectorsaregiveni ncartesianform usethescalarproductinsomegeometricalapplications Contents 1. Introduction 2. Definitionofthescalarproduct 3. Somepropertiesofthescalarproduct 3 4. Thescalarproductoftwovectorsgivenincartesianform 5 5. Someapplicationsofthescalarproduct 8 www.mathcentre.ac.uk 1 math centre2009
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1. Introduction Oneofthewaysinwhichtwovectorscanbecombinedisknownas the scalarproduct .When wecalculatethescalarproductoftwovectorstheresult,as thenamesuggestsisascalar,rather

thanavector. Inthisunityouwilllearnhowtocalculatethescalarproduc tandmeetsomegeometricalappli- cations. 2. Definition of the scalar product Studythetwovectors and drawninFigure1.Notethatwehavedrawnthetwovectorsso thattheirtailsareatthesamepoint.Theanglebetweenthet wovectorshasbeenlabelled Figure1.Twovectors, and ,drawnsothattheanglebetweenthemis Wedefinethescalarproductof and asfollows: Key Point The scalarproduct of and isdefinedtobe || cos where isthemodulus,ormagnitudeof isthemodulusof ,and istheanglebetween and Notethatthesymbolforthescalarproductisthedot

,andsowesometimesrefertothescalar productasthedotproduct.Eithernamewilldo. www.mathcentre.ac.uk 2 math centre2009
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Example Considerthetwovectors and showninFigure2. Suppose hasmodulus4units, has modulus5units,andtheanglebetweenthemis 60 ,asshown. 60 Figure2. and havelengths4and5unitsrespectively;theanglebetweenth emis 60 Wecanusethedefinitiongivenabovetofindthescalarproduct of and || cos = 4 cos 60 = 4 = 10 Sothescalarproductofthesevectorsisthenumber10.Notet hattheanswerisascalar,that isanumber,ratherthanavector. So,wehavelearntamethodo fcombiningtwovectorsto

produceascalar. 3. Some properties of the scalar product Commutativityanddistributivity Supposeforthetwovectorsinthepreviousexamplewecalcul atetheproductinadifferentorder. Thatis,supposewewanttofind .Thedefinitionof is || cos PerformingthecalculationusingthenumbersintheExample wefind || cos = 5 cos 60 = 5 = 10 So,weseethattheresultisthesamewhicheverwayaroundwep erformthecalculation. This istrueingeneral: Thispropertyofthescalarproductisknownas commutativity . Wepointitoutbecausein anotherunityoucanlearnaboutanotherwayofcombiningvec torsknownasthe vectorproduct

Thevectorproductisnotcommutativesotheorderinwhichwe writedownthetwovectorswill beveryimportant. www.mathcentre.ac.uk 3 math centre2009
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Key Point Thescalarproductis commutative .Thismeans Anotherpropertyofthescalarproductisthatitis distributiveoveraddition .Thismeansthat ) = Althoughweshallnotprovethisresulthereweshalluseitla teronwhenwedevelopanalternative formulaforfindingthescalarproduct. Key Point Thescalarproductis distributiveoveraddition .Thismeans ) = andalso,equivalently Thescalarproductoftwoperpendicularvectors Example Considerthetwovectors and

showninFigure3.Theanglebetweenthemis 90 ,asshown. Figure3.Theanglebetween and is 90 www.mathcentre.ac.uk 4 math centre2009
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Wecanusethedefinitiontofindthescalarproductof and || cos || cos 90 = 0 because cos 90 = 0 . Thisistruewhateverthelengthsof and . Sothescalarproductof twovectorswhichareatright-anglesisalways0. Wesaythat suchvectorsare perpendicular or orthogonal Key Point Fortwoperpendicularvectors = 0 Theconverseofthisstatementisalsotrue: ifwehavetwonon -zerovectors and andwe findthattheirscalarproductiszero,itfollowsthatthesev ectorsmustbeperpendicular.

We canusethisfacttotestwhethertwovectorsareperpendicul ar,asweshallseeshortly. 4. The scalar product of two vectors given in cartesian form Wenowconsiderhowtofindthescalarproductoftwovectorswh enthesevectorsaregivenin cartesianform,forexampleas = 3 + 7 and + 4 where and areunitvectorsinthedirectionsofthe and axesrespectively. Firstofallweneedtodevelopafewresultsinthefollowinge xamples. Example Supposewewanttofind .ThesevectorsareshowninFigure4. Notethatbecause and liealongthe and axestheymustbeperpendicular. So,fromthe resultwehavejustestablished,thescalarproduct mustbezero.

Forthesamereason,we www.mathcentre.ac.uk 5 math centre2009
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obtainthesameresultifwecalculate and Figure4.Theunitvectors and areperpendicular. Example Supposewewanttofind .ReferagaintoFigure4.Thevector isaunitvector,soitslength is1unit.Theanglebetweenavectoranditselfmustbezero.S || cos 0 = 1 = 1 since cos 0 = 1 Forthesamereason = 1 and = 1 Key Point If and areunitvectorsinthedirectionsofthe and axes,then = 0 = 0 = 0 = 1 = 1 = 1 Wecanusetheseresultstodevelopaformulaforfindingthesc alarproductoftwovectorsgiven incartesianform: www.mathcentre.ac.uk 6 math centre2009


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Suppose and then = ( NowfromthepreviousKeyPointmostofthesetermsarezero. T hosethatarenotsimplify because = 1 .Weobtain Thisistheformulawhichwecanusetocalculateascalarprod uctwhenwearegiventhecartesian componentsofthetwovectors. Key Point If and then Notethatausefulwaytorememberthisis: multiplythe componentstogether,multiplythe componentstogether,multiplythe componentstogether,andfinally,addtheresults. On occasionsyoumayseethisformreferredtoasthe innerproduct ofthevectors and . In thecontextofvectorsthissimplymeansthesumoftheproduc tsofthecorrespondingvector components.

Example Supposewewishtofindthescalarproductofthetwovectors = 4 +3 +7 and = 2 +5 +4 Theresultwehavejustderivedtellsustomultiplythe componentstogether,multiplythe componentstogether,multiplythe componentstogether,andfinallyaddtheresults.So = (4)(2) + (3)(5) + (7)(4) = 8 + 15 + 28 = 51 www.mathcentre.ac.uk 7 math centre2009
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Example Supposewewishtofindthescalarproductofthetwovectors +3 11 and = 12 +4 Notethatthe componentof iszero.So = ( 6)(12) + (3)(0) + ( 11)(4) 72 + 0 44 116 Itisoftenusefultomakeuseofcolumnvectornotation. Cons ideragainthelastexample. Writing

and ascolumnvectors 11 12 thescalarproductbecomes 11 12 = ( 6)(12) + (3)(0) + ( 11)(4) = 116 Exercises1. 1.If = 4 + 9 and = 3 + 2 find(a) ,(b) ,(c) ,(d) 2.Findthescalarproductofthevectors and 3.If = 4 + 3 + 2 and = 2 + 11 find(a) ,(b) ,(c) ,(d) 4.If = 3 + 2 + 8 showthat 5.If and 14 find(a) ,(b) ,(c) ,(d) 6.PointsA,B,andChavecoordinates(3,2,1),(5,4,2),and 1) respectively.Findthe scalarproductof AB and AC 5. Some applications of the scalar product Inthissectionwewilllookatsomewaysinwhichthescalarpr oductcanbeused. Usingthescalarproducttotestwhethertwovectorsareperp endicular

Acommonapplicationofthescalarproductistotestwhether twovectorsareperpendicular. Fromthedefinition || cos Ifboththevectors and arenon-zeroandwefindthat = 0 thenwecandeduce cos mustbezero,sothat = 90 ,i.e. and areperpendicular. Considerthefollowingexample. www.mathcentre.ac.uk 8 math centre2009
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Example Supposewewishtotestwhetherornotthevectors and areperpendicular,where Notethatneither nor iszero.Calculatingthescalarproductwefind = (3)(1) + (2)( 2) + ( 1)( 1) = 3 4 + 1 = 0 andso and areindeedperpendicular. Key Point If and arenon-zerovectorsforwhich = 0 ,then

and areperpendicular. Usingthescalarproducttofindtheanglebetweentwovectors Oneofthecommonapplicationsofthescalarproductistofind theanglebetweentwovectors whentheyareexpressedincartesianform. Fromthedefinitionofthescalarproduct || cos Wecanrearrangethistoobtainanexpressionfor cos cos || (1) Ifwearegiven and incartesianformwecanusetheresultobtainedinSection4t ocalculate .Wecanalsocalculatethemodulusofeachof and since and WiththisinformationEquation(1)canbeusedtofindtheangl ebetweenthetwovectors. www.mathcentre.ac.uk 9 math centre2009
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Key Point Theangle,

,betweenthetwovectors and canbefoundfrom cos || Example Supposewewishtofindtheanglebetweenthevectors = 4 + 3 + 7 and = 2 + 5 + 4 Thescalarproduct, ,hasalreadybeencalculatedonpage7andfoundtobe51. Themodulusofeachvectorisfound: + 3 + 7 74 + 5 + 4 45 Then,fromEquation(1), cos || 51 74 45 = 0 8838 (4sigfig) Finally,usingacalculator = cos 8838 = 27 90 Sotheanglebetweenthevectors and is 27 90 Findingthecomponentofavectorinthedirectionofanother vector Anotherapplicationofthescalarproductistofindthecompo nentofonevectorinthedirection ofanother. ConsiderFigure5.Wecanthinkofthevector

beingmadeupofacomponentinthedirection of ,( OA ),togetherwithaperpendicularcomponent,( AB ). Thecomponentinthedirection of isthe projection of onto Figure5.Theprojectionof onto is www.mathcentre.ac.uk 10 math centre2009
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Fromtherightangledtriangle OAB andusingtrigonometryweseethat cos andtherefore cos Usingtheformulafor cos obtainedinthepreviousapplicationwehave cos || Thiscanbewritteninthealternativeform Sotheprojectionof onto canbefoundbytakingthescalarproductof andaunitvector inthedirectionof ,i.e. a Key Point Theprojection, ,of onto canbefoundbytakingthescalarproductof

andaunitvector inthedirectionof a Example Supposewewishtofindthecomponentof = 3 + 4 inthedirectionof Aunitvectorinthedirectionof is Then (3 1 + 4) = = 2 Sothecomponentof = 3 + 4 inthedirectionof is www.mathcentre.ac.uk 11 math centre2009
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Exercises2. 1.PointsA,B,andChavecoordinates(3,2,1),(5,4,2),and 1) respectively. The scalarproductof AB and AC hasbeenfoundinthepreviousExercisesQ6.Findtheangle between AB and AC 2.Determinewhetherornotthevectors + 4 and areperpendicular. 3.Evaluate where = 4 + 8 .Hencefindtheanglethat makeswiththe axis.

4.Findthecomponentofthevector + 3 inthedirectionof = 7 + 2 AnswerstoExercises Exercises1 1.(a)30,(b)30,(c)97,(d)13. 2.0. 3.(a)27,(b)27,(c)29,(d)126. 4.Bothequal77. 5.(a) 50 ,(b) 50 ,(c)29,(d)201. 6. 14 Exercises2. 1. 131 2. Theirscalarproductiszero. Theyarenon-zerovectors. W ededucethattheymustbe perpendicular. 3. = 4 .Therequiredangleis 63 4. 15 57 = 1 987 (3d.p.). www.mathcentre.ac.uk 12 math centre2009