Medan Elektromagnetik, Slides of Electronics

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SUWARNOSUWARNO

ELECTROMAGNETICS IELECTROMAGNETICS I

EL 3001

Vector & Coordinate systems

EL 3001

Vector & Coordinate systems

School of Electrical Engineering and Informatics INSTITUT TEKNOLOGI BANDUNG 2006

VektorVektor

Vector quantity : expressed by its magnitude and

direction. A A a A

A

A A a A

A

A

a A 

Unit Vector

VECTOR ANDVECTOR AND VECTOR COMPONENTSVECTOR COMPONENTS

CARTESIAN

2 2 2

ax ay a z A

a^ A x y z

A x y^ z A A A

A A A  

   

A 2 2 2 A   Ax  Ay  Az

VECTOR ADDITIONVECTOR ADDITION

C= A + B=B+A

VECTOR SUBSTRACTIONVECTOR SUBSTRACTION

C= A-B= A +(- B)

B

C=A-B

A

-B

VECTOR PRODUCTVECTOR PRODUCT

dot product: the product of magnitude of the two vectors and cosine of the angle between the them.

A.B=B.A =AB cos AB

Projection A on B

Length= A cos AB

2

B

A. B

B

Vektor : cos

B B

A  AB 

DOT PRODUCTDOT PRODUCT



 





 

 

 

  



 A B

A. B cos

Anglebetween two vectors

A.A A

A.(B C) A.B A.C( )

A.B B.A( )

1 AB

(^22)



A

distributi ve

commutativ e

CROSS PRODUCTCROSS PRODUCT

x y z

x y z

x y z

B B B

A A A

a a a

AxA



  



  

 

AxB

a xa a ;a xa a ;a xa a

0

Ax(B C) AxB AxC (distribut ive)

AxB BxA (anticommut ative)

x y z y z x z x y

TRIPPLE PRODUCTTRIPPLE PRODUCT

x y z

x y z

x y z

C C C

B B B

A A A

C

C C

A.(Bx )

Ax(B.C) no meaning

A.(BxC) meaningful l

Sequenceisimportant!!

A.(Bx ) B.( xA) C.(AxB)

B(A.C)- C(A.B)

Ax(BxC) (AxB) xC

22

2 a 3 a 3 a Unit vectorA a

a.Length A 2 3 3 22 x y z A

2 2 2    

   

50 , 2^ o 22

cos^3 A

A. a cos

b.Anglebetween Adan Y axis

• 1 y (^1)   

^  

^  ^  

 ^  

   

145 , 1^ o 22 27

cos A B

A. B

cos

c.Anglebetween A and B

  -^1  ^1  ^  

OP sin ( 180 ) 2 , 68

d.DistancefromO to B

3  ^  

A o

COORDINATE SYSTEMS

• Cartesian (x,y,z)

• Cylindrical (z)

• Spherical (r,)

CYLINDRICALCYLINDRICAL

CYLINDRICAL ELEMENTSCYLINDRICAL ELEMENTS