EUM 103 Chapter 2.1 Multi Variables Functions

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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