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VectorsI

2.2 Vectors&Scalars

AVectorhasonlyTwoproperties…MagnitudeandDirection.

That’saweirderconceptthanyouthink.AVectordoesnotnecessarilystartatagivenpoint,butcan“float”about,butstillbetheSAMEvector.

Avectorisusuallydenotedinbold,likevectora,orsometimesitisdenoted𝑎",ormanyotherdeviationsexistinvarioustextbooks.

𝐴𝐵%%%%%⃗ = 𝒂= 𝑎~=~𝑎

Thetildeunderisthewaythistextbookshowsavector,butitshardtodoinWord,soIwillusejustaboldletter!

AVectorisdrawnwithaheadandtail,toconveydirection.(Ihopeyoucanworkoutwhichendiswhich)

UnderstandingtheGraphicalrepresentationofvectorsisimperative:

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Additionofvectors:

Considerthefollowinggeometricrepresentationofvectoraddition.

Looksprettyweird/abstract.Canyoushowmoreclearlytherepresentationof𝒂 + 𝒃

Whenweaddvectors“geometrically”wesimplyplacethemheadtotail.TheresultantVectorisdrawnfromthe“tail”ofthefirstvector,tothe“head”ofthesecondvector.Wecanevenaddmultiplevectorstogether.

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Doesthislookbetter?

Itisimportantthatyouuseyoureyestohelpyourbrainseetheseoperationsaseasy.

Mathsisaboutseeingthroughthecomplexity,andlookingatthingsstrategicallytomakeitmoreeasytounderstand.

NotethatIusedtheoriginofaCartesianplaneasmystartpoint.Althoughvectorsdonothaveastartpoint,sometimesthismakesiteasierto“see”whatishappening!

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TASK:

Draw2vectors,𝒂and𝒃anywhereonyourpage.

Now;

Geometricallyadd𝒂 + 𝒃

Geometricallyadd𝒃 + 𝒂

Whatisyourhypothesisabouttheorderinwhichyouaddvectors?

Takealookatthis“Applet”…https://mathinsight.org/applet/vector_sum

TASK:

Drawanyvectorandlabelit𝒂.

Now,geometricallydrawvector–𝒂.

Now,drawanothervector–𝒂somewhereelseonyourpage

Whatisthedifferencebetweenyour2vectorsof–𝒂?

TASK:

Drawanyvectoronyourpageandlabelit𝒄.

Now;

Trytodrawavectorthatrepresents2𝒄.

TASK:

Drawanyvectoronyourpageandlabelit𝒅.

Now;

IfIsaythatVectordivisionisnotdefined,thinkaboutwhatyoucoulddotothisvectortomakeitsmaller,butwithoutchangingitsdirection?

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So,youwereright…youcanaddandsubtractvectorsinanyorder.Itdoesn’tmatter.YoucanrepresentavectorgeometricallyANYWHEREonyourpageasitdoesnothaveaStartpoint.

YoucanalsoSCALARmultiplyavector.

AndiftheScalarislessthan1,thenthevectorgetssmaller.

Letsjustdoublechecksomethings:

Canyouseethatvector𝒅isEXACTLYtheSAMEVectoras𝒃

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HowaboutaScalarofavector.

Althoughwedenotethisoperationas𝑘𝒂1(kisaconstant)tryNOTtothinkofitasMultiplication.

Whatisascalar?…it’stheenlargement,orreductionofthemagnitudeofsomething.

Ascalemodelcar,issupposedtobeanexactreplicaofarealcar,just“shrunkdown”.

ThewordScalarcomesupagainlaterandweNeedtothinkofScalardifferentlythanMultiplication…?

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Remember,vectorscanrepresentanythingwithmagnitudeanddirection,sowecanhavevectorsfordisplacement,velocity,forceandalthoughthischapterwillinitiallymainlystaywithinthe2Dspace,weneedtopushintovectorsin3DJ.

Haveaplaywith:

https://www.physicsclassroom.com/Physics-Interactives/Vectors-and-Projectiles/Vector-Addition/Vector-Addition-Interactive

Wearesearchingfortextbookquestionsinthebooknottodo…J

GodoExercise2.2.

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2.3.Positionvectorsintheplane

Itisconvenienttorepresentvectorsbycomponentsinthe𝑥and𝑦directionandthe𝑥directionisdenotedwithan8̂andtheydirectionisdenotedwitha:̂.

(thethird“Cartesian”dimensionwecall𝑧,andinvectorsthisalignsto𝒌=)

In2DCartesianform:𝒂 = 𝑥𝚤̂ + 𝑦𝚥̂

In3DCartesianform:𝒂 = 𝑥𝚤̂ + 𝑦𝚥̂ + 𝑧𝑘@

Sometimesweabbreviateitevenfurtherandwecansimplyputtheminparenthesisorpointybrackets.

In2DCartesianform:𝒂 = (𝑥, 𝑦) = ⟨𝑥, 𝑦⟩

In3DCartesianform:𝒂 = (𝑥, 𝑦, 𝑧) = ⟨𝑥, 𝑦, 𝑧⟩

Andcanalsobeputintocolumn(matrix)form;

In2DColumnform:𝒂 = F𝑥𝑦G

In3Dcolumnform:𝒂 = H𝑥𝑦𝑧I

AndcanalsobeputintoPolarform;

In2DPolarform:𝒂 = [𝑟, 𝜃](risthemagnitudeandThetaistheangle)

Vectorscanbescalarmultiplied,addedandsubtracted.Asyouhaveseenwecandothisgeometrically,buttheseoperationscanalsobeperformedmathematically.

Butfirst,letsstickwith2Dvectorsandworkouthowweget𝑥and𝑦.HereweuseTrigonometry.

Becausevectorshaveamagnitudeanddirection,wecouldsimplysayavectoris10unitslongandhasanangleof30degrees.

**Allanglesaremeasuredfromthepositive𝑥axis,inananti-clockwisedirection.**

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OK,thinkYr9trigonometry…ThelengthofthevectoristheHypotenuse.Thesizeofthe𝑥componentusesthecosineratioandthelengthofthe𝑦componentusesthesineratio.It’sasimpleas:

𝑥 = vectorlength × cos 𝜃

𝑦 = vectorlength × sin 𝜃

Hangon,letstalkinmathematicalterms…J

ThelengthofthevectoriscalledMagnitude,andisdenotedby|𝒂|.

***caution,thisisthesamesymbolweusedforthedeterminantofamatrix,somakesureyoudon’tgetconfused!

Soletssay

𝑥 = |𝒂| cos 𝜃

𝑦 = |𝒂| sin 𝜃

TERMINOLOGY:

Trouble,wenotonlyneedtoknowthemagnitudeofavectorisdenotedby|𝒂|butitisalsodenotedas𝑟.(fromPolarform)

Sowecouldalsosay

𝑥 = 𝑟 cos 𝜃

𝑦 = 𝑟 sin 𝜃

Sowereallyhave;

𝑥 = |𝒂| cos 𝜃 = 𝑟 cos 𝜃

𝑦 = |𝒂| sin 𝜃 = 𝑟 sin 𝜃

J

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Whatifwehaveavectorincomponentformandwewanttoknowhowbigitisandwhatdirectionitisacting?

UsingPythagoras,wehave|𝒂| = 𝑟 = [𝑥\ + 𝑦\

𝜃 = tan]^𝑦𝑥

Whatifwehavethevector’smagnitudeanddirectionandwewanttoputitintocomponentform?

𝑥 = |𝒂| cos 𝜃 = 𝑟 cos 𝜃

𝑦 = |𝒂| sin 𝜃 = 𝑟 sin 𝜃

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OK…letsputitalltogether.Asdifferenttextbookssaythingsdifferently,therewillbemanydifferentrepresentationsofthesameconcept.Letslookatjustafew:

In2D:

𝑂𝑃"""" = 𝑎a = 𝒂 = 𝑥𝚤̂ + 𝑦𝚥̂ = |𝑎| cos 𝜃 �̂� + |𝑎| sin 𝜃 𝚥̂ = ⟨𝑥, 𝑦⟩ = F𝑥𝑦G = [𝑟, 𝜃],

WehaveaMagnitude,|𝑎| = 𝑟 = [𝑥\ + 𝑦\

AndthedirectionoftheVectorisgivenby𝜃,where tan 𝜃 = cd

In3D:

GivenVector𝑂𝑃"""" = 𝑎a = 𝒂 = 𝑥𝚤̂ + 𝑦𝚥̂ + 𝑧𝑘@ = ⟨𝑥, 𝑦, 𝑧⟩,

|𝑎| = [𝑥\ + 𝑦\ + 𝑧\

OK,haveyounoticedthatthe𝑖, 𝑗&𝑘arewearing“hats”.ThehattellsusthatthevectorisaspecialvectorcalledaUnitVector.WeneedtotalkaboutUnitvectors!

AunitvectorisavectorofMagnitude1unit.

Thinkaboutastick10unitslong,togetthatstick1unitlong,wewouldscalaritby ^

^h!Vectorsarethesame.Weneedto“Scaleouroriginalvector,eitherbigger,

orsmallersothatitsmagnitudeis1.Hencewehave;

𝑎i =1|𝑎| 𝒂

Andweshallsimplifyittoreadas;

𝑎i = 𝒂|𝑎| =

𝒂𝑟

Now,specificallythinkingabout𝑖, 𝑗&𝑘,theseareUnitVectorsinthedirectionofthe𝑥, 𝑦&𝑧axisrespectfully.So,hopefullyyoucan“see”howtherepresentationworks…

𝒂 = 𝑥�̂� + 𝑦𝚥̂

Thevector𝒂isactuallythesameasthegeometricalrepresentationofthesumofthetwovectorsrepresentedby𝑥𝚤̂and𝑦𝚥̂…J

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Somethingfurthertoconfuseyou.

ThereisadifferencebetweenaPositionVector,andaVector.

APositionVectordescribesaPositionintheplane,or“Point”,whereasaVectorisseenasa“line”thathasmagnitudeanddirection(butnostartingpoint).

LetsdoaSimpleexample…ConsiderthePositionVectors⟨0,1⟩and⟨2,3⟩…hangon,don’tjustconsiderthem,plotthemonaCartesianplane.

ThesetwoPositionVectorsdescribeaVectorthatjoinsthese2points.CanyouworkouttheVectorthatjoinsthese2points?

Theanswer,𝒗 = 2�̂� + 2𝚥̂

Makesureyoucanseehowthisworks…itisimportant…(wewilllikelyneedtodothisontheboard)!!!

𝐴𝐵%%%%%⃗ = 𝒃 − 𝒂

Addingandsubtractingvectorsisaseasyas:

Given𝒂 = 3𝚤̂ − 4𝚥̂ + 3𝑘@ ,and𝒃 = 2𝚤̂ + 3𝚥̂ + 2𝑘@

𝒂 + 𝒃 = (3 + 2)𝚤̂ + (−4 + 3)𝚥̂ + (3 + 2)𝑘@ = 5𝚤̂ − 𝚥̂ + 5𝑘@

Thetaismeasuredanticlockwisefromthe𝑥-axis…thisisdifferentthantruebearings…youwillneedtomanuallyconvertbetweenthetwoforms!

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TASK:

OnaCartesianplane,plotthecoordinates(1, 1), (3, 1), and(2, 3).

Whatshapeisthis?

WecouldalsothinkoftheseasPositionVectors,asPositionVectorsrepresent“Points”,soIcouldhavesaid,plotthePositionVectors⟨1, 1⟩, ⟨3, 1⟩and⟨2, 3⟩…andyouwouldhavedrawntheexactsamediagram.

HopefullyyousaidTriangle,andspecificallyanIsoscelesTriangle.Notonlydothreepointsrepresentatriangle,butatrianglealsohas3sides.RatherthandescribingthetrianglewithPositionVectors(points),wecanalsodescribeatrianglewiththreeVectors.

TASK:

Giventhethreepointsoftheprevioustask,useyourthoroughunderstandingofhowtogeometricallyarrangevectors,tocomeupwiththe3vectorsthatdescribethetriangleinthepreviousTask.

LetsallocatethePositionVectorsas:

𝑨 = ⟨1, 1⟩,𝑩 = ⟨3, 1⟩,𝑪 = ⟨2, 3⟩

andaskyoutofindVectors𝑨𝑩,𝑩𝑪and𝑨𝑪thatdescribethatsametriangle.

Answertotask:

𝑨𝑩 = −𝑨 + 𝑩 = ⟨−1,−1⟩ + ⟨2, 3⟩ = ⟨1, 2⟩

𝑩𝑪 = −𝐁 + 𝐂 = ⟨−2,−3⟩ + ⟨3, 1⟩ = ⟨1, −2⟩

𝑨𝑪 = −𝑨 + 𝑪 = ⟨−1,−1⟩ + ⟨3, 1⟩ = ⟨2, 0⟩

GoDoExercise2.3.

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2.4.Vectorproblemsolving

Notawholeheapofnewknowledgehere,justworkingouthowtosolveproblems!

Thismaybeintuitive,butasamathematicalstatementwecansay…If,

𝒂 = 𝑘𝒃

then𝒂and𝒃areparallel.

ThischapterwilltakesomepersistenceanddeterminationJ

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2.5TheDotProduct(ScalarProduct)

AdotproductisaScalarValue(thenamegivesthataway)thatistheresultofanoperationoftwovectorswiththesamenumberofcomponents.ItistheSumoftheProductsofeachrespectivecomponents.

In2D:𝒂 ⋅ 𝒃 = (𝑥^ × 𝑥\) + (𝑦^ × 𝑦\)

Italsoworksin3D:𝒂 ⋅ 𝒃 = (𝑥^ × 𝑥\) + (𝑦^ × 𝑦\) + (𝑧^ × 𝑧\)

Geometricallyspeaking,theDotProductreferstomultiplyingthemagnitudeofonevector,withthemagnitudeoftheComponentoftheothervectorthatisinthesamedirectionasit.Sowehaveanotherformulaforit.

𝒂 ⋅ 𝒃 = |𝒂||𝒃| cos 𝜃

Thisoneformulaworksin2Dand3Dsituations!

Becauseofthisdefinitionwecansaythatingeneraltermsthescalaranswertellsushow‘alike’onevectoristoanother,soifouranswerisZero,thentheyareNotatallalike,orwecansaytheyareperpendiculartoeachotherandiftheDotProductisnegative,wecansaytheyareinoppositedirections.

Linkingthetwoformulaswehave:

𝒂 ⋅ 𝒃 = |𝒂||𝒃| cos 𝜃 = 𝑥^𝑥\ + 𝑦^𝑦\ + 𝑧^𝑧\

WeusethisruletoFindtheanglebetweentwovectorsin2Dand3D.

TheDotProductisagoodwaytoseeifvectorsareperpendicular,becausecos 90 = 0,thedotproductwillbeequaltoZero.

Note:youmeasuretheanglebetween2vectorsbyplacingthemtailtotail!

DoExercise2.5

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Ifyoulookontheinterweb,youmayfindmanyconfusingexplanationsofthedotproduct.ThiswebsiteisasclearasIcanfindanddoesanadequatejobattryingtoexplainit!

http://mathinsight.org/image/dot_product_projection

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2.6.Resolutes

Therearethreeformula’storemember…

Remember:

ScalarResoluteof𝒂on𝒃isgivenby 𝒂 ∙ 𝒃=

VectorResoluteof𝒂parallelto𝒃is 𝒂∥ = z𝒂 ∙ 𝒃={𝒃=

VectorResoluteof𝒂perpendicularto𝒃is 𝒂| = 𝒂 − z𝒂 ∙ 𝒃={𝒃=

***Note:thewayIwritethescalarresoluteisdifferentthanthebook.BywritingitMYway,italignstothewayyousayit.Theresoluteof𝒂on𝒃…sotheacomesfirst***

Done,godotheexercisequestions…

Hangon…letsteachforUnderstanding!

Understandinghowitworksfromfirstprincipleswillhelpyousolveproblems!

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TheScalarResolute:

Definition:Thescalarresoluteofvector𝒂onvector𝒃isthemagnitudeofthecomponentofvector𝒂actinginthedirectionofvector𝒃.

Geometricallyusingyear9trigonometryit’sthevalue…|𝑎| cos 𝜃;

Thisishowto“see”theScalarresoluteofvectorAonVectorB.

ItissimplytheMagnitudeofthecomponentofVectorAinthedirectionofvectorB.

It’sassimpleasthat.

Hmmm…that’sdifferentthantheformulaonthelastpage?

That’sbecause|𝑎|andcos 𝜃maybedifficulttofind;

Didyounoticethat|𝑎| cos 𝜃showsupinthedotproduct,well‘nearly’;

Thedotproductruleis;

𝒂 ⋅ 𝒃 = |𝒂||𝒃| cos 𝜃

Letsdividebothsidesby|𝑏|

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𝒂 ⋅ 𝒃|𝒃| =

|𝒂| cos 𝜃

Thereforewecansay;

𝒂 ⋅ 𝒃|𝒃| = scalarresolute

Letsneatenthatupabit

𝒂 ⋅𝒃|𝒃| = scalarresolute

Andas 𝒃|𝒃| = 𝐛� ,weget;

Thescalarresoluteofaontobis…𝒂 ⋅ 𝒃=

Rememberthatthedotproductof2vectors(asabove),givesusascalaranswer(anumber),sothatmakessense,becausethescalarresolute(asthenamesuggests)isasinglenumbervalueJ

RecallthatthescalarresoluteistheMagnitudeofthecomponentofonevectorthat’sinthedirectionoftheothervector.

Thinkabout2forcevectors…thescalarresoluteishowmuchtheforceofonevectoris“helping”theothervectorJ

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TheparallelVectorResolute:

Simply,theParallelVectorResoluteisScalarResolute,butasanactualvector!

Icanusetheexactsamediagram…

TheVectorresolute(parallel)ofA,ontoB,issimplythevectorcomponentofA,inthedirectionofB.

Justputan‘arrow’ontothat|𝐴| cos 𝜃line!

SoifwehaveaUnitVectorforB,wesimplymultiplyitbythescalarresolute…andweendupwithavectorinthedirectionofBandwiththemagnitudeofthescalarresolute!

|𝐴| cos 𝜃𝐵@ **Note:intheabove,𝐵@isaUnitVECTOR.

Orinformulaformwehave:𝒂∥ = z𝒂 ∙ 𝒃={𝒃=

Hence,theresultofthisoperationisaVector.Ascalarmultipliedbyaunitvector,isaVector!

Specifically,itisthe“vectorcomponent”ofAinthedirectionofB.

VectorResolute…asthenameimplies,youranswerhastobeaVECTOR.

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ThePerpendicularVectorResolute:

TASK:

Drawthesamediagraminyourbook.

DrawthedottedlineinRed(withanarrowatthetop).

ThisisthecomponentofAthatisperpendiculartoB.

CanyoushowGeometrically,thatthiscanberepresentedbyvectorAminustheparallelvectorresolute,or;

Wecanshowthisgeometricalrepresentationmathematicallyas;

𝑨 − |𝐴| cos 𝜃𝐵@

orinformulaform:𝒂| = 𝒂 − z𝒂 ∙ 𝒃={𝒃=

andwearrivebackatourthreeresoluteformula;

ScalarResoluteof𝒂on𝒃isgivenby 𝒂 ∙ 𝒃=

VectorResoluteof𝒂parallelto𝒃is 𝒂∥ = z𝒂 ∙ 𝒃={𝒃=

VectorResoluteof𝒂perpendicularto𝒃is 𝒂| = 𝒂 − z𝒂 ∙ 𝒃={𝒃=

DoExercise2.6