eum 103 chapter 2.1 multi variables functions
TRANSCRIPT
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Intwodimensionalalgebraic,vectorisrepresentedby
x andy components
Magnitudeofthevector
Angle betweenvectorandxaxis
Inthreedimensionalalgebraic,vectorisrepresented
byx ,y andz components
Magnitudeofthevector
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Directionangles:
Anglebetweenvectorandaxis:
Example
Solution
Calculatethedistancesofthepointsi)(1,0,2),ii)( 2,1, 3)from
theorigin.Calculatethedistancebetweenthesetwopoints
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zerovectorandunitvector
Zerovector:0
Unitvector: magnitude 1andrepresentedby
Fortwodimensional,theunitvectors havingthe
directionsofx andy axis.
Ifwehaveanyvectorv=(vx,vy),wecanwriteitintocomponentform:
Intwodimensional,
Similartothetwodimensional,unitvectorsparalleltothex,y andz
axisare
where
Thus,
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Example
Solution
If isavector,findtheunitvectorwhichhasthesamedirectionasthatofv
Additionandscalarmultiplicationvector
Undergeometricvectoraddition breakthevectors
intox andycomponents
Combinethex andycomponentsseparately
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Ifu=(ux,uy),v=(vx,vy)and isascalar
Inalgebraicproperty,if[u,v&w]arevectorsand[ & ]
arescalarsand0isthezerovector
(i)DotProduct
Thedotproduct(multiply) oftwovectors
toobtainascalar:
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Lengthofavector:
Anglebetweentwovectors:
PropertiesofDotProduct:
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(ii)CrossProduct
Thecrossproductusmultiplyoftwovectorstoobtaina
vectorandonly definedfor3dimensionalspace.
Lets
Let betheangle
between vectors v
andu,andassumethat0<
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PropertiesofCrossProduct:
whenthetwovectorsuandvareorthogonal
,
theproducts oftheircorrespondingelementsis0(vi) Ifu andv areparallel,theanglebetweenthem
either0or180degrees,thenvxv=0
Fortriplescalarproducts:
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Example
Findthecrossproductofuv,ifu=(2,4, 5)andv=(3, 2,1)
Example
Ifa=i j 2k,b=2i+j+3kthenfindaxbwithunitvector
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(ii)VectorGradient
Thegradientofafunctionf=f(x,y)isdefined:
whichwillpointsinthedirectionofthegreatestrateof
changeofthefunctionfgreatestrateofchange
Example
Findthegradientoff(x,y)=x2 y+3y