Fall 2006Fall 2006

Scalar Quantity(mass, speed, voltage, current and power)

1- Real number (one variable)

2- Complex number (two variables)

Vector Algebra(velocity, electric field and magnetic field)

Magnitude & direction

1- Cartesian (rectangular)2- Cylindrical3- Spherical

Specified by one of the following coordinates best applied to application:

Page 101

Conventions

• Vector quantities denoted as or v• We will use column format vectors:

• Each vector is defined with respect to a set of basis vectors (which define a co-ordinate system).

v

Tvvvvvv

v

v

v

321321

3

2

1

v

Unit vector

Magnitude

222

222

ˆˆˆˆ

ˆˆˆ

zyx

zyx

zyx

zyx

AAA

AzAyAx

A

Aa

AAAAA

AzAyAxA

Page 101-102

a

Vectors• Vectors represent directions

– points represent positions

• Both are meaningless without reference to a coordinate system– vectors require a set of basis

vectors– points require an origin and a

vector space

y

x

y

x

v

v

v

vPv

both vectors equal

Co-ordinate Systems• Until now you have probably used a Cartesian

basis:– basis vectors are mutually orthogonal and

unit length– basis vectors named x, y and z

• We need to define the relationship between the 3 vectors.

5.0

7.0

5.0

v

Cartesian Coordinate System

Vector Magnitude• The magnitude or norm of a vector of

dimension n is given by the standard Euclidean distance metric:

• For example:

• Vectors of length 1(unit) are normal vectors.

222

21 nvvv v

11131

1

3

1222

Normal Vectors

• When we wish to describe direction we use normalized vectors.

• We often need to normalize a vector:

vv

vv

222

21

1

nvvv

Vector Addition and SubtractionVector Addition and Subtraction

)(ˆ)(ˆ)(ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

zzyyxx

zyx

zyx

BAzBAyBAxBbAaBACcC

BzByBxBbB

AzAyAxAaA

xC yC zC

Addition

Subtraction

212

212

21212

12121212

122112

)()()(

)(ˆ)(ˆ)(ˆ

zzyyxxR

zzzyyyxxxR

RRPPR

R12

Page 103

Vector Addition

u+v

y

x

v

u

Vector Addition

• Addition of vectors follows the parallelogram law in 2D and the parallelepiped law in higher dimensions:

vuvu

9

7

5

6

5

4

3

2

1

Vector Subtraction

y

x

v

u v-u

Problem 3-1

• Vector starts at point (1,-1,-2) and ends

at point (2,-1,0). Find a unit vector in the

direction of .

A

A

y

x

z

20ˆ11ˆ12ˆ zyxA

Problem 3-1

20ˆ11ˆ12ˆ zyxA

y

x

z

2,1,11 P

0,1,22 P

A

20ˆ11ˆ12ˆ zyxA

2ˆˆ zxA

24.2541 A

89.0ˆ45.0ˆ24.2

2ˆˆˆ zx

zx

A

Aa

Vector MultiplicationVector Multiplication1- Simple Product2- Scalar or Dot Product3- Vector or Cross Product

1- Simple Product

)(ˆ)(ˆ)(ˆˆ zyx kAzkAykAxkAaAkB

Scalar or Dot Product

ABABBA cos

Page 104

Vector Multiplication by a Scalar

• Each vector has an associated length

• Multiplication by a scalar scales the vectors length appropriately (but does not affect direction):

Dot Product

• Dot product (inner product) is defined as:

332211

3

2

1

321 vuvuvu

v

v

v

uuuT

vuvu

i

iivuvu

cosvuvu

Dot Product

• Note:

• Therefore we can redefine magnitude in terms of the dot-product operator:

• Dot product operator is commutative & distributive.

223

22

21 uuu uuu

uuu

Problem 3.2

• Given vectors:

– show that is perpendicular to both and .

A

B

C

zyxA ˆ3ˆ2ˆ

3ˆˆ2ˆ zyxB

2ˆ2ˆ4ˆ zyxC

Recall

10cosˆˆ

10cosˆˆ

10cosˆˆ

zz

yy

xx

zz

yy

xx

0ˆˆˆˆˆˆ zyzxyx

Also, recall

Similarly

0 BA

If the angle between the two vectors is 90.

Problem 3.2

02682ˆ2ˆ4ˆˆ3ˆ2ˆ zyxzyxCA

06282ˆ2ˆ4ˆ3ˆˆ2ˆ zyxzyxCB

ABABBA cos

0ˆˆˆˆˆˆ

1ˆˆˆˆˆˆ

)ˆˆˆ()ˆˆˆ(

xzzyyx

zzyyxx

BzByBxAzAyAxBA zyxzyx

zzyyxx BABABABA AAAA

BA

BAAB

1cos

CABACBA

ABBA

)(

Commutative property

Distributive property

A

Page 105

Vector or Cross Product

ABABnBA sinˆ

)(ˆ)(ˆ)(ˆ xyyxzxxzyzzy BABAzBABAyBABAxBA

0ˆˆˆˆˆˆ

ˆˆˆˆˆˆˆˆˆ

)ˆˆˆ()ˆˆˆ(

zzyyxx

yxzxzyzyx

BzByBxAzAyAxBA zyxzyx

Page 105-106

)(ˆ)(ˆ)(ˆ xyyxzxxzyzzy BABAzBABAyBABAxBA

Show

zyx AAA zyxA ˆˆˆ

zyx BBB zyxB ˆˆˆ

yxz

xzy

zyx

ˆˆˆ

ˆˆˆ

ˆˆˆ

yzx

xyz

zxy

ˆˆˆ

ˆˆˆ

ˆˆˆ

Recall

xyyxzxxzyzzy

yzxzzyxyzxyx

zyxzyx

BABABABABABA

BABABABABABA

BBBAAA

zyx

xyxzyz

zyxzyxBA

ˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

)(ˆ)(ˆ)(ˆ xyyxzxxzyzzy BABAzBABAyBABAxBA

Problem 3.3

• In the Cartesian coordinate system, the three corners of a triangle are P1(0,2,2), P2(2,-2,2), and P3(1,1,-2). Find the area of the triangle.

y

x

z

y

x

z

y

x

z

y

x

z 2,2,01P

y

x

z 2,2,01P

y

x

z 2,2,01P

y

x

z 2,2,01P

y

x

z 2,2,01P

y

x

z 2,2,01P

2,2,22P

y

x

z 2,2,01P

2,2,22P

y

x

z 2,2,01P

2,2,22P

y

x

z 2,2,01P

2,2,22P

y

x

z 2,2,01P

2,2,22P

y

x

z 2,2,01P

2,2,22P

2,1,13 P

y

x

z 2,2,01P

2,2,22P

2,1,13 P

Let

4ˆ2ˆ21 yxB PP

and

4ˆˆˆ31 zyxC PP

represent two sides of the triangle.

Since the magnitude of the cross product is the area of the parallelogram, half of this magnitude is the area of the triangle.

9182

1324

2

1464256

2

12816

2

1

ˆ2ˆ8162

116ˆ4ˆ8ˆ2

2

1

ˆˆ16ˆˆ4ˆˆ4ˆˆ8ˆˆ2ˆˆ22

1

4ˆˆˆ4ˆ2ˆ2

1

2

1

222

zyzyz

zyyyxyzxyxxx

zyxyxCB

xx

A

0 z y z 0 x

• The area of the triangle is 9 sq. units.

Scalar and Vector Triple ProductsScalar and Vector Triple Products

??

??

???

?

CBACBACBA

CBACBACBA

CBACBACBA

CBACBACBA

zyx

zyx

zyx

CCC

BBB

AAA

BACACBCBA

)()()( BACCABCBA

Scalar Triple Product

Vector Triple Product

Page 107-108

222

222

ˆˆˆˆ

ˆˆˆ

zyx

zyx

zyx

zyx

AAA

AzAyAx

A

Aa

AAAAA

AzAyAxA

Addition

Subtraction

Vector MultiplicationVector Multiplication

1- Simple Product2- Scalar or Dot Product3- Vector or Cross Product

Scalar Triple Product

Vector Triple Product

Orthogonal Coordinate Systems: (coordinates mutually perpendicular)

Spherical Coordinates

Cylindrical Coordinates

Cartesian CoordinatesP (x,y,z)

P (r, Θ, Φ)

P (r, Θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Page 108

-Parabolic Cylindrical Coordinates (u,v,z)-Paraboloidal Coordinates (u, v, Φ)-Elliptic Cylindrical Coordinates (u, v, z)-Prolate Spheroidal Coordinates (ξ, η, φ)-Oblate Spheroidal Coordinates (ξ, η, φ)-Bipolar Coordinates (u,v,z)-Toroidal Coordinates (u, v, Φ)-Conical Coordinates (λ, μ, ν)-Confocal Ellipsoidal Coordinate (λ, μ, ν)-Confocal Paraboloidal Coordinate (λ, μ, ν)

Cartesian CoordinatesP(x,y,z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(r, θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

forward

Cartesian Coordinates

zyx AzAyAxA ˆˆˆ

Page 109

x

y

z

Z plane

y planex plane

222zyx AAAAAA

xyz

x1

y1

z1

Ax

Ay

Az

( x, y, z)Vector representation

Magnitude of A

Position vector A

),,( 111 zyxA

111 ˆˆˆ zzyyxx

Base vector properties

0ˆˆˆˆˆˆ

1ˆˆˆˆˆˆ

xzzyyx

zzyyxx

yxz

xzy

zyx

ˆˆ

ˆˆˆ

ˆˆˆ

Back

x

y

z

Ax

Ay

AzA

B

Dot product:

zzyyxx BABABABA

Cross product:

zyx

zyx

BBB

AAA

zyx

BA

ˆˆˆ

Back

Cartesian Coordinates

Page 108

Cartesian Coordinates

Differential quantities:

Length:

Area:

Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysd

dydzxsd

z

y

x

ˆ

ˆ

ˆ

dxdydzdv

Page 109Back

BaseVectors

A1

r radial distance in x-y plane

Φ azimuth angle measured from the positive x-axis

Z

r0

20

z

Cylindrical Coordinates

ˆˆˆ,ˆˆˆ,ˆˆˆ rzrzzr

zr AzAArAaA ˆˆˆˆ

Pages 109-112Back

( r, θ, z)

Vector representation

222zr AAAAAA

Magnitude of A

Position vector A

Base vector properties

11 ˆˆ zzrr

Dot product:

zzrr BABABABA

Cross product:

zr

zr

BBB

AAA

zr

BA

ˆˆˆ

B A

Back

Cylindrical Coordinates

Pages 109-111

Cylindrical Coordinates

Differential quantities:

Length:

Area:

Volume:

dzzrddrrld ˆˆˆ

rdrdzsd

drdzsd

dzrdrsd

z

r

ˆ

ˆ

ˆ

dzrdrddv

Pages 109-112Back

ˆˆˆ,ˆˆˆ,ˆˆˆ RRR

Spherical Coordinates

Pages 113-115Back

(R, θ, Φ)

AAARA Rˆˆˆ

Vector representation

222 AAAAAA R

Magnitude of A

Position vector A

1ˆRR

Base vector properties

Dot product:

BABABABA RR

Cross product:

BBB

AAA

R

BA

R

R

ˆˆˆ

Back

B A

Spherical Coordinates

Pages 113-114

Spherical Coordinates

Differential quantities:

Length:

Area:

Volume:

dRRddRR

dldldlRld R

sinˆˆˆ

ˆˆˆ

RdRddldlsd

dRdRdldlsd

ddRRdldlRsd

R

R

R

ˆˆ

sinˆˆ

sinˆˆ 2

ddRdRdv sin2

dRdl

Rddl

dRdlR

sin

Pages 113-115Back

zz

yx

yxr

ˆˆ

cosˆsinˆˆ

sinˆcosˆˆ

zz

yx

yxr

AA

AAA

AAA

cossin

sincos

Back

Cartesian to Cylindrical Transformation

zz

xy

yxr

)/(tan 1

22

Page 115

y

x

z

Convert the coordinates of P1(1,2,0) from the Cartesian to the Cylindrical and Spherical coordinates.

0,2,11P

y

x

z

Convert the coordinates of P1(1,2,0) from the Cartesian to the Cylindrical coordinates.

0,2,11P

r

51422 yxr

radx

y107.1

1

2tantan 11

0h

y

x

z

0,4.63,24.2

0,107.1,51

radP

r

51422 yxr

radx

y107.1

1

2tantan 11

0h

y

x

z

4.63,90,24.2

107.1,57.1,51

radradP

r

5014222 zyxr

radx

y107.1

1

2tantan 11

0h

radz

yx57.1

0

5tantan 1

221

Convert the coordinates of P1(1,2,0) from the Cartesian to the Spherical coordinates.

y

x

z

Convert the coordinates of P3(1,1,2) from the Cartesian to the Cylindrical and Spherical coordinates.

2,1,11P

21122 yxr

2h

radx

785.01

1tan

1tan 11

y

x

z

Convert the coordinates of P3(1,1,2) from the Cartesian to the Cylindrical coordinates.

2,4

,23 radP

21122 yxr

2h

radx

785.01

1tan

1tan 11

y

x

z

Convert the coordinates of P3(1,1,2) from the Cartesian to the Spherical coordinates.

45,3.35,45.23P

6211 222222 zyxr

rad616.02

2tan

2

11tan 1

221

rad4

1tan1

1tan 11