01a vector analysis

8/13/2019 01a Vector Analysis

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VECTOR ANALYSIS


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VECTORS

Has magnitude

Has directiondirectiondirectiondirection

Can be represented graphically by arrowsarrowsarrowsarrows

Vector addition of vectors AAAA and BBBB C = A + BC = A + BC = A + BC = A + B (1.1)

Figure 1.1 Triangle law of vectorFigure 1.1 Triangle law of vectorFigure 1.1 Triangle law of vectorFigure 1.1 Triangle law of vectoraddition.addition.addition.addition.

( )1.1BACrrr

+=


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VECTORS

Vector addition is commutative

C = A + B = B + AC = A + B = B + AC = A + B = B + AC = A + B = B + A, (1.2)

( )1.2ABBACrrrrr

+=+=

Figure 1.2 Parallelogram law ofFigure 1.2 Parallelogram law ofFigure 1.2 Parallelogram law ofFigure 1.2 Parallelogram law of

vector addition.vector addition.vector addition.vector addition.


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VECTORS

Vector addition is associative

Figure 1.3 Vector additionFigure 1.3 Vector additionFigure 1.3 Vector additionFigure 1.3 Vector addition

Is associative.Is associative.Is associative.Is associative.

D =D =D =D = A + BA + BA + BA + B + C,+ C,+ C,+ C,

A + B = EA + B = EA + B = EA + B = E....

D =D =D =D = EEEE + C.+ C.+ C.+ C.

B + C = FB + C = FB + C = FB + C = F....

D = A +D = A +D = A +D = A + F.F.F.F.

( A + BA + BA + BA + B) + C = A + () + C = A + () + C = A + () + C = A + (B + CB + CB + CB + C).).).).


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VECTORS

Example

Figure 1.4 Equilibrium of forces:Figure 1.4 Equilibrium of forces:Figure 1.4 Equilibrium of forces:Figure 1.4 Equilibrium of forces: FFFF1111 + F+ F+ F+ F2222 ==== FFFF3333


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VECTORS

Vector subtraction, same as adding a vector of

reverse direction

AAAA ---- B = A + (B = A + (B = A + (B = A + (----B).B).B).B).

A = EA = EA = EA = E ---- B.B.B.B.


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VECTORS

Vector AAAA can be represented by coordinatescoordinatescoordinatescoordinates

Origin, (0,0,0)Origin, (0,0,0)Origin, (0,0,0)Origin, (0,0,0) = start of vector arrow AAAA

(AAAAxxxx , A, A, A, Ayyyy ,,,, AAAAzzzz ) =) =) =) = coordinates of the endpoint of AAAA


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VECTORS( ) (1.3)zy,x,r

r

r = directed line segment from origin

to tip of arrow; parallel to vector A

Figure 1.5 Cartesian components and direction cosines of A.Figure 1.5 Cartesian components and direction cosines of A.Figure 1.5 Cartesian components and direction cosines of A.Figure 1.5 Cartesian components and direction cosines of A.


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VECTORS Direction cosines

( ) ( )

A

rAr

r

r

r

r

andAbetweenangle

,andbetween xangle,cos,coscos

cos

x=

===

==

x

x

Ax

A

A

r

x


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VECTORS Direction cosines

( ) ( )

A

rAr

r

r

r

r

andbetweenangle

,andbetweenangle,cos,coscos

cos

y

y

y

A

yAy

A

A

r

y

=

=

==

==


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VECTORS

A

A

r

zA

r

yA

r

x zyx ====== coscoscos

( )

1.4.1coszcosycos rrrx ===

. .coscoscos zyx ===

( )1.4.31coscoscos 222 =++

( ) ( )1.5AAA1.4.3,and1.4.2,1.4.1,From

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z

2

y

2

x ++=A

The magnitude of AAAA is proportional to rrrr through some scale:

e.g. 1cm: 1 Newton


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VECTORS

Coordinate representation of vector AAAA

(1.6)A zyx AAA ++=r

r

zyx

unit vectorsunit vectorsunit vectorsunit vectors

,, zyx

( )( )

( ) axis-zalongrunit vecto1,0,0

axis-yalongrunit vecto0,1,0axis-xalongrunit vecto0,0,1

=

==

z

yx


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VECTORS

Vector addition using coordinates

( ) ( ) ( ) (1.7)A zzyyxx BABABA ++= zyxBvr


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VECTORS - ROTATION OF THE COORDINATE AXES

Figure 1.6 Rotation of Cartesian coordinate axes aboutFigure 1.6 Rotation of Cartesian coordinate axes aboutFigure 1.6 Rotation of Cartesian coordinate axes aboutFigure 1.6 Rotation of Cartesian coordinate axes about

the zthe zthe zthe z----axis.axis.axis.axis.


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cossin'

(1.8)sincos'

yxy

yxx

+=

+=


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Transformation of coordinates of vector AAAA by

rotation of coordinate axes

(1.9)sincos' yxx AAA +=

Form invariance (or covariance) cossin' yxy AAA +=

( ) ( ) 2122221222 /zyx/

zyx A'A'A'AAA ++=++=A


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Change notation

cossin'

(1.9)sincos'

yxy

yxx

AAA

AAA

+=

+=

cossin'(1.8)sincos'

yxyyxx+=

+=

( )

cossin

1.11sin,cos

2221

1211

==

==

aa

aa

( )

2221212

2121111

'

1.12'

equationstionTransforma

xaxax

xaxax

+=

+=


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( )

2221212

2121111

'

1.12'


xaxax

xaxax

+=

+=

aaaa ,,,:tscoefficientionTransforma

jiij xxa and'betweenangleofcosinecosine,direction=

( )

( ) ( ) ( )1.13sincos,'cos

sin,'cos

21221

2112

=+==

==

xxa

xxa

( )1.14.2,1'2

1

===

ixaxj

jiji


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( )1.15,....2,1'

systemldimensionaNorfour,Three,

1

NiVaVN

j

jiji ===


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and'betweenangleofcosineSince jiij xxa =

( )1.14.2,1'2

1

===

ixaxj

jiji( )

2221212

2121111

'

1.12'


xaxax

xaxax

+=

+=

( )1.16a'j

iij

xxa

=

( )

( )1.16b'

or'

-rotationInverse

2

1

ij

i

j

j

iijj ax

xxax =

=

=


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( )

1.15,....2,1'

systemldimensionaNorfour,Three,

1

== =

N

j

jiji NiVaV

( )

(1.16b)From(1.16a)From

1.17'

''

11

=

=

==

N

j

j

i

jN

j

j

j

ii V

x

xV

x

xV


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( )

( )1.19

1.18

forconditionityOrthogonal

jkkiji

jki

ikij

ij

aa

aa

a

=

=

( )

kj

kj

jk

jk

=

==

for0

1.20for1

deltaKronecker

k

j

k

i

i

j

i

k

i

j

x

x

x

x

x

x

x

x

x

x

=

=

'

'''

:Proof

ii


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01a vector analysis

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