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VectorsYear12Term1

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Vectors-AVectorhasTwoproperties…MagnitudeandDirection

-Avectorisusuallydenotedinbold,likevectora,or𝑎 ⃑,ormanyothers.

In2D

-𝒂 = 𝑥𝚤 + 𝑦𝚥

-𝒂 = 𝑥,𝑦

-where,𝑥 = 𝑎 cos 𝜃and𝑦 = 𝑎 sin 𝜃

In3D

-𝒂 = 𝑥𝚤 + 𝑦𝚥 + 𝑧𝑘

-𝒂 = 𝑥,𝑦, 𝑧

Recall:

-𝚤 , 𝚥 & 𝑘areunitvectors(length1)inthedirectionsoftheirrespectiveaxis

-𝑎 = !!

-YoushouldknowhowtoAdd,SubtractandScalarmultiplyvectors

-NotethatScalarMultiplicationreferstotheenlargementorcontractionofavector…(nottobeconfusedwithScalarProductwedolater)

-YouneedtobeabletodotheseoperationsbothMathematicallyandGeometrically.

-Magnitudeofavectorisdenoted𝑂𝑃 = 𝒂 = 𝒓 = 𝑥! + 𝑦! + 𝑧!

-in2D,theanglebetweenvectorsisgivenby𝜃,wheretan𝜃 = !!

-in3D,usethedotproducttofindtheanglebetween2vectors.

TASK

Given𝒂 = 2, 1, 3 and𝒃 = 1,−1, 4

CalculatethemagnitudeoftheresultantVectorcreatedby𝑎 − 2𝑏.

Thenderiveaunitvectorinthedirectionof𝑎 − 2𝑏.

Thenfindaunitvectorintheoppositedirection.

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Ans: 34, 0, !!", !!!"and 0, !!

!", !!"

TheDotProduct(ScalarProduct)

AdotproductisaScalarValuethatistheresultofanoperationoftwovectorswiththesamenumberofcomponents.ItistheSumoftheProductsofeachcomponent.

𝑎 ⋅ 𝑏 = 𝑥!×𝑥! + 𝑦!×𝑦! + 𝑧!×𝑧!

ThereisalsoaGeometricrelationshipdefinedbytheDotProduct.

𝑎 ⋅ 𝑏 = 𝑎 𝑏 𝑐𝑜𝑠𝜃

Therefore,wecansay:

𝑎 ⋅ 𝑏 = 𝑎 𝑏 𝑐𝑜𝑠𝜃 = 𝑥!×𝑥! + 𝑦!×𝑦! + 𝑧!×𝑧!

WeusethisruletoFindtheanglebetweentwovectorsinboth2Dand3D.

TheDotProductisagoodwaytoseeifvectorsareperpendicular,becausecos 90 = 0,thedotproductwillbeequaltoZero.

𝒂 ⊥ 𝒃 𝑖𝑓 𝒂 ∙ 𝒃 = 0

TASK

Whatistheanglebetweenvector 1, 1 and 2,−2 ?DrawthesevectorsonaCartesianplanetoconfirmyouranswerisreasonable.

Ans:90

TextBook,Exercise3A:Thisisrevision,sodoingthemALLwilltakeyounotimeatall.

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TheVectorProductin2D

Justastwopointscandefineastraightline,twoVectorsisonewaytodescribeaPlane.

Task:Pickup2pensandvisualisetheonlyPlanethatcontainsthesetwo“vectors”.

TheVectorProduct/CrossProductofanytwovectorsproducesavectoratanormaltotheplanecreatedbyVectors𝒂and𝒃.

ThereisaweirdprocessforthisintheTextonpage104.Ihaveneverusedthiswaybefore,soIwillnotgothroughit,butwillskiptoaquicker,easierandmoredetailedprocessusingMatrices.(asperpage105)

If 𝒂 = 𝑎!,𝑎!,𝑎! and 𝒃 = 𝑏!, 𝑏!, 𝑏!

Then

𝒂 × 𝒃 =𝑖 𝑗 𝑘𝑎! 𝑎! 𝑎!𝑏! 𝑏! 𝑏!

RecalltheDeterminantofa3x3Matrix?

𝒂 × 𝒃 =𝑎! 𝑎!𝑏! 𝑏!

𝑖 −𝑎! 𝑎!𝑏! 𝑏!

𝑗 +𝑎! 𝑎!𝑏! 𝑏!

𝑘

andthisResultantVectorwillbeataNormaltobothinitialvectors,andassuchwillalsobeataNormaltothePlanetheydefine.

**thereisNoneedtoknowthe2ndvectorproductrule𝑎×𝑏 = 𝑎 𝑏 sin𝜃 𝑛,becausethereislimitedapplicationofit.YouwillseeIhaveNotaskedyoutodoQ1**

TASK

Findavectorthatisperpendiculartoboth 2, 0, 0 and 0, 1, 0 .

Ans: 0, 0, 2

Consideryouranswer.Doesitseemreasonable?Visualiseyourinitialtwovectors.Wheredotheylie?Whatistherelationshipbetweenallthreevectors?

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AnimportantelementoftheVectorProductisthattheModulus(Magnitude/Length)ofaVectorProductgivestheareaoftheParallelogramcreatedbythetwovectors.

So, 𝐴𝑟𝑒𝑎 𝑃𝑎𝑟𝑎𝑙𝑙𝑜𝑔𝑟𝑎𝑚 𝒂𝒃 = 𝒂×𝒃

And,byextrapolation,theareaofatrianglecreatedbythetwovectorscanbecalculateas;

𝐴𝑟𝑒𝑎 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝒂𝒃 = 𝒂×𝒃!

Recall 𝑂𝑃 = 𝒂 = 𝑟 = 𝑥! + 𝑦! + 𝑧!

TASK

DrawaCartesianPlane.

Byplottingvectors 0, 2 and 3, 0 youhaveaParallelogram.UseaGrade9techniquetofindtheareaofthisspecialparallelogram.

Ans:6

NowcalculatetheVectorProduct(CrossProduct)ofthetwoinitialvectors,andfindthemodulusoftheresultantvector?

Hint:changetheinitialvectorstohavevaluesfor𝑖, 𝑗 & 𝑘first.

Ans:6

Hopefullyyouarenotsurprisedatgettingthesameanswer…J

**Care**Foreverysetofvectors,thereareTWONormalvectors(eachverticallyoppositetoeachother).Ensureyouunderstandthe“rightytighty–leftyloosey”conceptofwhichdirectiontheNormalvectorgoesin.Refertopage103oftheTextbook.

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TASK

Onmywhiteboard,if𝑖isverticaland𝑗ishorizontal,determine 2, 0, 0 × 0, 5, 0 anddescribeitsdirection.

Ans: 0, 0, 10 ,outofthewall.

TASK

DetermineaunitvectorintheoppositedirectionofyourresultinthepreviousTask.

Ans: 0, 0,−1

TASK

Ithinkallofthesevectorsareperpendicular?Whatoperationcanwedotoconfirmthatalloftheseareinfactperpendiculartoeachother?

Ans:Dotproduct,

andcorrect,alltheDotproductsare0(acceptforthedotproductofthetwovectorsintheoppositedirectionofeachother).

WedoNOTneedtolearnanythingtodowithForceorTorque,asthisschooldoesnotdotheDynamicschapter(bummer!)SothisisabitofanEasylesson!

TextBook,Exercise3B

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TheTripleProducts

Therearetwotripleproducts,theScalarTripleProduct,andtheTripleVectorProduct.WeareonlyconcernedwiththeScalarTripleProductatthisyearlevel.

TheScalarTripleProduct:asthenamesuggests(scalar),givesusasinglenumberanswer,whichisthevolumeofaparallelepipedformedby𝒂,𝒃&𝒄.

ScalarTripleProduct…𝒂 ⋅ (𝒃×𝒄)

Asyoucansee,weneedtoperformthevectorproductfirst,sothatourdotproductwillgiveusascalaranswer.

TASK

Givenvectors,𝒂 = 1, 1, 2 ,𝒃 = 2,−1, 1 and𝒄 = 1, 2,−1 ,performthefollowingTripleProducts.

𝑎 ⋅ (𝑏×𝑐) (𝑎×𝑏) ⋅ 𝑐 𝑏 ⋅ (𝑎×𝑐) 𝑏 ⋅ (𝑐×𝑎)

Ans:12,12,-12,12

Fromyourresult,whatisyourhypothesisabouttheScalarTripleProductorderofoperations?Andwhatdoyounoticeaboutthemodulusofyoursolutions?

TheScalarTripleProduct,(justlikewhenwemultiply3sidestogetheronarectangle),willgiveusaVolume!Tobeprecise,theModulusoftheScalarTripleProductwillgiveustheVolumeoftheParallelepiped,formedbyvectors𝒂,𝒃and𝒄.

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TASK

LookbackatyourworkingforthepreviousTask.WhenyoucalculatedtheDotproduct,whatdidyoumultiply𝑎!by?

Ifyoucan’tseeit,I’llhelpyou.

1× −1×−1− 1×2 …andinsidethosebrackets,itlooksalotlike(ad-bc)…J

Ans:𝑎!× 𝑏!𝑐! − 𝑏!𝑐!

So,whatwehaveseenisthattheScalarTripleProductissimplytheDeterminantofthe3x3matrixformedbythe3vectors.

SowecansaythescalarTripleProductforvectors𝒂,𝒃and𝒄is:

𝑎! 𝑎! 𝑎!𝑏! 𝑏! 𝑏!𝑐! 𝑐! 𝑐!

TASK

UsingtheDeterminantmethod,findthevolumeofaparallelepipedformedbythethreevectors,𝒂 = 1, 1, 2 ,𝒃 = 2,−1, 1 and𝒄 = 1, 2,−1 .

Ans:12

TextBook,Exercise3C

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EquationstoPlanes

Feelfreetoreadpage116ofyourtexttogiveyouastartingpoint.Ithinkthatexplanationisalittleconfusing,soIwillgiveyouanalternative:

Describingaplaneisbestdonebyreferringtoallsuchvectorsthatarenormaltoagivenvector,andthatpassthroughsomefixedpoint.

TASK:Givethissomethought.Don’tmerelyacceptit,questionhowandwhywewouldwanttodefineaplaneinthisway?

Here,wewillsay 𝑟 =apositionvectortoANYpointintheplane

𝑛 =apositionvectorNORMALtotheplane

𝑎 =apositionvectorofaSETpointintheplane

Wecanseethat𝑟 − 𝑎liesparalleltotheplane.(makesureyouCANseethis)

Knowingthatthedotproductoftwovectorsis0iftheyareatanormaltoeachother,wecannowsay,

𝑟 − 𝑎 ⋅ 𝑛 = 0

or 𝑟 ⋅ 𝑛 − 𝑎 ⋅ 𝑛 = 0

andwearriveataVectorEquationforaPlane:

𝑟 ⋅ 𝑛 = 𝑎 ⋅ 𝑛

OK….𝑟,𝑎 and 𝑛arePositionVectors.Specifically,𝑟isthevectorthatdescribesANYpointintheplane,soitisa“General”vector,sowecansay:

𝑟 = 𝑥,𝑦, 𝑧

andletsset

𝑛 = ∝,𝛽, 𝛾

Nowjustlookingatthelefthandside,𝑟 ⋅ 𝑛becomes, 𝑥,𝑦, 𝑧 ⋅ 𝛼,𝛽, 𝛾

Or, 𝛼𝑥 + 𝛽𝑦 + 𝛾𝑧

Andontherighthandside,𝑎 ⋅ 𝑛istheDotproductoftwovectors,soitissimplyaconstant,where𝑎and𝑛areknown,andthedotproductisascalar,hencewearriveatwhatiscalledthe:

EquationtoaPlaninCartesianForm 𝛼𝑥 + 𝛽𝑦 + 𝛾𝑧 = 𝑑

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TASK

Letsstarteasy…Considertheverticalplanethatisadistanceof2inthe𝑥direction.

Clearlyavectoratanormaltothiscouldbe𝑛 = 2, 0, 0

Clearlyapositionvectorthatliesintheplanecouldbe𝑎 = 2, 2, 2

Whatistheequationtotheplane?

Ans:𝑟 ∙ 2, 0, 0 = 4

TASK

Considertheexactsameplane.Showthatpoint 2,−1, 3 liesintheplane.

Ans: 2,−1, 3 ∙ 2, 0, 0 = 4

Satisfiesequation,∴liesintheplane

TASK

Find3otherpointsintheplaneandPROVEtheylieintheplane.

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TASK

Considerthesameplane.Whathappensif𝑛doesn’t“connect”totheplane?

Clearlythepositionvector 4, 0, 0 isalsoatanormaltotheplane.

Giventhis,whatisarevisedequationtotheplane?

Ans:𝑟 ∙ 4, 0, 0 = 8

TASK

Re-proveyourabovepointsstillsatisfythisequation.

What’syourHypothesis?

Clearly,𝑛doesnotneedtotouch/endattheplane,itsimplyhastobeataNormaltotheplane…

and

Iamhopingyouhavehypothesizedthat;justas;

4𝑥 − 2𝑦 + 6 = 0isthesameas𝑦 = 2𝑥 + 3;

wecanhaveplaneequationsthatmaylookdifferent,butareactuallythesamePlane!(scalar’sof)

TASK

Createamoredifficultplane,andre-performtheprevious5Taskswithappropriatepoints.

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TASK

Returntoourniceeasyverticalplane.Wehavetried𝑛 = 2, 0, 0 and𝑛 =4, 0, 0 ,andthisshowedthatsolongas𝑛isaNormalVector,thePlaneequationwillwork.Nowletsconsiderwhathappenswhenweset𝑛tobeaUnitVector.

Clearly𝑛 = 1, 0, 0

PuttheequationtotheplaneinCartesianform𝛼𝑥 + 𝛽𝑦 + 𝛾𝑧 = 𝑑

Ans:𝑥 + 0𝑦 + 0𝑧 = 2

TASK

Whatdoes𝑑representwithregardstotheplane?

Ans:thedistanceoftheplanetotheOrigin

**Important,whenwehavevaluesoftheunitvectorofn,or𝑛,then,𝑎 ⋅ 𝑛isthePerpendicularDistancefromtheOrigin,totheplane,hencethisistheshortestdistancefromtheOrigintothePlane.

ThinkbacktoYr9mathsandlinearfunctions...

Twopointscandescribeastraightline…butwecanalsodescribeastraightlinebyitsslopeandonepointontheline.

Similarlywecandescribeaplaneinmorethanonewayaswell.

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TASK:Considercarefullyandexplainthedifferencebetween:

Twovectorscandefineaplane,andthreepositionvectorscandefineaplane.

TASK:Whatdoesouraboveprocessrelyontocreateanequationtoaplan,apositionvector,oravector?

TASK:Howdowetranslate3positionvectors,to2vectors?

Doyourbesttodraw3positionvectorsin3Dandconvertthesetotwovectorsthatlieintheplane.

Drawingasetofaxesin3Dishard…eventhetextbookstruggles(asperthediagramonpage116)…butdoyourbestJ

Hint:ifthethreepointsarea,bandc,thentwovectorsintheplanewouldbec-aandc-b.

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TASK

Given3pointsdescribedbythepositionvectors,

𝒂 = 3𝚤 + 4𝚥 + 3𝑘,𝒃 = 2𝚤 + 3𝚥 + 2𝑘and𝒄 = 2𝚤 − 𝚥 + 𝑘,

orIcouldwritethisas𝒂 = 3,4,3 ,𝒃 = 2,3,2 and𝒄 = 2,−1,1 .

DerivetheequationtotheplanethatcontainsallthreepointsinbothVectorFormaswellasCartesianForm.

Hint,firstfindtwovectorsthatlieintheplane…findaNormaltobothofthem.FindtheVectorequationfirst,thentheCartesianfollowsquitedirectly.

Ans: 𝑟 ⋅ −3,−1,4 = −1,or−3𝑥 − 𝑦 + 4𝑧 = −1

TextBookExercise3D,