Cartesian components of vectors mcTYcartesian AnyvectormaybeexpressedinCartesiancomponentsbyusin gunitvectorsinthedirectionsof thecoordinateaxes
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Cartesian components of vectors mcTYcartesian AnyvectormaybeexpressedinCartesiancomponentsbyusin gunitvectorsinthedirectionsof thecoordinateaxes

Inthisunitwedescribetheseunitvect orsintwodimensionsandinthree dimensionsandshowhowtheycanbeusedincalculations Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthis

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Cartesian components of vectors mcTYcartesian AnyvectormaybeexpressedinCartesiancomponentsbyusin gunitvectorsinthedirectionsof thecoordinateaxes




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Presentation on theme: "Cartesian components of vectors mcTYcartesian AnyvectormaybeexpressedinCartesiancomponentsbyusin gunitvectorsinthedirectionsof thecoordinateaxes"— Presentation transcript:


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Cartesian components of vectors mc-TY-cartesian1-2009-1 AnyvectormaybeexpressedinCartesiancomponents,byusin gunitvectorsinthedirectionsof thecoordinateaxes. Inthisunitwedescribetheseunitvect orsintwodimensionsandinthree dimensions,andshowhowtheycanbeusedincalculations. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: identifythecoordinateunitvectorsintwodimensionsandi nthreedimensions;

determinewhetherasetofcoordinateaxesinthreedimensio nsislabelledasaright-handed system; expressthepositionvectorofapointintermsofthecoordin ateunitvectors,andasa columnvector; calculatethelengthofapositionvector,andtheanglebetw eenapositionvectoranda coordinateaxis; writedownaunitvectorinthesamedirectionasagivenposit ionvector; expressavectorbetweentwopointsintermsofthecoordinat eunitvectors. Contents 1. Vectorsintwodimensions 2 2. Vectorsinthreedimensions 3 3. Thelengthofapositionvector 5 4. Theanglebetweenapositionvectorandanaxis 6 5. Anexample www.mathcentre.ac.uk 1 math centre2009


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1. Vectors in two dimensions Thenaturalwaytodescribethepositionofanypointistouse Cartesiancoordinates. Intwo dimensions,wehaveadiagramlikethis,withan -axisanda -axis,andanorigin .Toinclude vectorsinthisdiagram,wehaveavector associatedwiththe -axisandavector associated withthe -axis. (3, 4) Ifwetakeanypointinthisdiagram,forinstancethepoint withcoordinates (3 4) ,thenwe canwrite OP = 3 + 4 Itisimportanttoappreciatethedifferencebetweenthesetw oexpressions. Thenumbers (3 4) representasetofcoordinates,referringtothepoint . Buttheexpression + 4 isavector, thepositionvector OP

.Analternativewayofwritingthisisasacolumnvector: meansthesameas + 4 Sometimesonenotationisused,andsometimestheother. Key Point Intwodimensions,theunitvectorsinthedirectionsofthet wocoordinateaxesarewrittenas and . Ifapoint hascoordinates x, y thenthepositionvector OP maybewrittenasa combinationoftheseunitvectors, OP orequivalentlyasacolumnvector OP www.mathcentre.ac.uk 2 math centre2009
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2. Vectors in three dimensions Inthreedimensionswehavethreeaxes,traditionallylabel led and ,allatrightanglesto eachother. Anypoint cannowbedescribedbythreenumbers,thecoordinateswithr espect

tothethreeaxes. Nowtheremightbeotherwaysoflabellingtheaxes. Forinsta ncewemightinterchange and ,orinterchange and . Butthelabellinginthediagramisastandardone,anditisc alleda right-handedsystem Imaginearight-handedscrew,pointingalongthe -axis. Ifyoutightenthescrew,byturningit fromthepositive -axistowardsthepositive -axis,thenthescrewwillmovealongthe -axis. Thestandardsystemoflabellingisthatthedirectionofmov ementofthescrewshouldbethe positive direction. Thisworkswhicheveraxiswechoosetostartwith,solongasw egoroundthecycle andthenbackto again.Forinstance,ifwestartwiththepositive

-axis,thenturnthescrew towardsthepositive -axis,thenwelltightenthescrewinthedirectionofthepo sitive -axis. Key Point Aright-handedsystemisasetofthreeaxes,labelledsothat rotatingascrewfromthepositive -axistowardsthepositive -axiswilltightenthescrewinthedirectionofthepositive -axis. www.mathcentre.ac.uk 3 math centre2009
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Nowletstakeapoint inthree-dimensionalspace,withcoordinates x, y, z . Theposition vectorofthepointwillbethelinesegment OP ( , , Wecannowwrite OP where isaunitvectorinthedirectionofthe -axis. Againitisimportanttoappreciatethe difference. Thenumbers x, y, z

representasetofcoordinates,referringtothepoint . But theexpression isavector,thepositionvector OP .Wesometimeswritethisisasa columnvector: meansthesameas Key Point Inthreedimensions,theunitvectorsinthedirectionsofth ethreecoordinateaxesarewritten as and .Ifapoint hascoordinates x, y, z thenthepositionvector OP maybewritten asacombinationoftheseunitvectors, OP orequivalentlyasacolumnvector OP www.mathcentre.ac.uk 4 math centre2009
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3. The length of a position vector Whatisthelengthofthepositionvector OP Toanswerthisquestion,westartbydroppingaperpendicula rfrom downtothe x, y

-plane. Weshallcallthisnewpoint .Thenwejointhepoint uptothe and axes,againatright angles.Weshallcallthetwonewpoints and ( , , Nowweknowsomeofthelengthsinthisdiagram. First,thelen gth PQ istheheightofthe point abovethe x, y -plane.Sothatlengthmustbe Thelength OA isthedistancealongthe coordinateaxis,sothatlengthmustbe . Andthe length BQ isthesameasthelength OA ,sothatmustalsobe Inthesameway,thelength OB isthedistancealongthe coordinateaxis,sothatlengthmust be .Andthelength AQ isthesameasthelength OB ,sothatmustalsobe ( , , Nowwejointhepoints and .Then OAQ isaright-angledtriangle,andsois OBQ

.Sothe length OQ canbefoundbyusingPythagorassTheorem,ineitherofthes etriangles.Weobtain theformula OQ OA AQ (or OB BQ Nowwecanusetheright-angled triangle OQP . IfweapplyPythagorassTheoremtothis www.mathcentre.ac.uk 5 math centre2009
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triangle,weobtain OP OQ QP Key Point If isthepointwithcoordinates x, y, z thenthelength,ormagnitude,ofthepositionvector OP isgivenbytheformula OP OP 4. The angle between a position vector and an axis ( , , Nowwehavefoundthelengthoftheline OP :itis .Andwealsoknowthelength oftheline OA :itis . Butthetriangle POA

isaright-angledtriangle,sowecanwritedown thecosineoftheangle POA .Ifwecallthisangle ,then cos Thequantity cos isknownasa directioncosine ,becauseitisthecosineofananglewhichhelps tospecifythedirectionof istheanglethatthepositionvector OP makeswiththe -axis. www.mathcentre.ac.uk 6 math centre2009
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Ofcoursewecandothesameforthe -axisandforthe -axis.Weobtain cos where istheanglethat OP makeswiththe -axis,and cos where istheanglethat OP makeswiththe -axis. Key Point Thedirectioncosinesofthepoint describetheanglesbetweenthepositionvector OP and thethreeaxes.If hascoordinates x, y, z

thenthedirectioncosinesaregivenby cos cos cos Nowwecanfindaninterestingformulaifwetakethethreedire ctioncosines,squarethem,and addthem.Whatis cos + cos + cos Well, cos cos cos sothat cos + cos + cos = 1 Sothesquaresofthedirectioncosines,whenaddedtogether ,equal1.Whatusecouldthisbe? Infactittellsusthatthevector cos + cos + cos isaunitvector.Thatisbecausethemagnitudeofthisvector isthesquarerootofthequantity cos +cos +cos ,whichwehavejustseenisequalto1.Andthisvectorisalsoi nthesame directionasouroriginalvector OP ,becausethethreenumbers cos cos and cos areinthe sameratioas and

.Sowehavefoundaunitvectorinthedirectionofourorigina lposition vector OP www.mathcentre.ac.uk 7 math centre2009
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Key Point Thedirectioncosinesofanypoint satisfytheequation cos + cos + cos = 1 Aunitvectorinthesamedirectionasthepositionvector OP isgivenbytheexpression cos + cos + cos 5. An example Supposewehaveapoint withcoordinates (1 2) andanotherpoint withcoordinates (2 4) . Wecanthenformthevector AB . Nowwhatisthemagnitudeofthisvector,and whatareitsdirectioncosines? Wecananswerthesequestionsbywritingthetwopositionvec tors OA and OB intermsofthe unitvectors and .Weobtain OA + 2

OB = 2 + 4 So AB AO OB OB OA = (2 + 4 + 2 Nowwhenwesubtractexpressionsinvolvingtheunitvectors and wejustsubtractthe correspondingcomponentsseparately.So AB = (2 ) + ( ) + (4 + 2 www.mathcentre.ac.uk 8 math centre2009
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Withthisexpressionforthevector AB ,wecancalculateitsmagnitude.Itis AB AB + ( 1) + 2 Wecanalsocalculatethethreedirectioncosinesof AB .Theyare cos cos cos Exercises 1.Findthelengthsofeachofthefollowingvectors (a) + 4 + 3 (b) (c) (d) (e) (f) 2.Findtheanglesgivingthedirectioncosinesofthevector sinQuestion1. 3.Determinethevector AB foreachofthefollowingpairsofpoints

(a)A(3,7,2)andB(9,12,5) (b)A(4,1,0)andB(3,4,-2) (c)A(9,3,-2)andB(1,3,4) (d)A(0,1,2)andB(-2,1,2) (e)A(4,3,2)andB(10,9,8) 4.ForeachofthevectorsfoundinQuestion3,determineauni tvectorinthedirectionof AB Answers 1. (a) 29 (b) 30 (c) (d) 5 (e) 14 (f) 2. (a) 68 42 56 (b) 24 111 79 (c) 90 26 116 (d) 90 90 (e) 36 122 105 (f) 54 54 54 3. (a) + 5 + 3 (b) + 3 (c) + 6 (d) (e) + 6 + 6 4. (a) 70 (6 + 5 + 3 (b) 14 + 3 (c) 10 + 6 (d) (e) www.mathcentre.ac.uk 9 math centre2009