vectors i - weebly€¦ · on a cartesian plane, plot the coordinates (1,1),(3,1),and (2,3). what...
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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