t.2

CHAPTER 1. VECTOR ANALYSIS AND MAXWELL'S EQUATIONS IN INTEGRAL FORM

and OP.OQ = loPlloQlcos?

.'. cos9 = cosdcosc[, + cosBcospr + cosTcosTr

(a) B=QP

oP = QP+OQ

.'.8 = QP = OP-OQ| ^ ^ \ t^ \= [-/a] + za")-\zax +41, +Jaz/

= -2^,- 3a, -a.

(b) Assume the smaller angle between A and B is 9. The magnirude of projection of B on

is lBlcos0.

(c)

... A

'h, ^ A'B.'. |II|COSU = --:--- =

tAlt'^l

-5= -- =414

-1.336

B = lAllBlcoso

(a, +za. -:a.X-za, -:a" -a.)

-

.,!I'+2' -t(-3)rt.(-2) +2.(-3) + (-3)(-1)

-

114

-t.336

cosg=ry=

.'. 0 = 110.9"

"lt-zl' +(-3)'z+(-1)2

(d)

la- a.. a, I

| ' ' 'lAxB = lt 2 3llr

l-2 -3 -ll= (-2 -9)a,+(6+ 1)a, + (-3 + 4)a,

= -1la" +7ar+a,

lA x Bl = Jl l' +7\ l' = 13.017

.'. unit vector = ,A t B,

= -0.841a + 0.535a + 0.0765alAxBl r )

1.3 Atx= l,y=).,7={

.4.=a *4a +12ar_lz

unit vectorAIAIl^-l

=

a +4a +l2ar,yzr._.------i

"ll" +4'+12'0.0788a +03152a +0.9457arlz

L4 Atx=2,y=3A = 3a +2a +3aryz

B=4a +4a

(a)

(b)

A. B = (3x 4) + (2x 4)+ (3 x0) = 20

^ ABCOSU =:-:---: =

lAllBl= 0.7538

1.5

.'. I = 41.08"

(c) The projection of B along the direction of A is l8lcos0

lBlcosg = ^[4' + 4t .0.7538 = 4.264

If A, B, and C are perpendicular to each other, then

A.B=0 B.C=O A.C=0

CHAPTERI.vEcToRANALYSISANDMAXWELL'SEQUATIoNSININTEGRALFORM

1.6 (a)

A'C =

+

A'B =

B.C

BxC

(su, * za, + :a.)' (r a "

+ cra, + ^ ")

15+29+3 = 0

cr= -9

(sa,+zrr+:a.)'(4., * Zar+ B,a,)

58,+418,=O

(4u, * 2rr+ B,a,)'(r., - 9a, + a.)

34-18*8,=Q

+ B, = l4'5

B' = -25'5

lu' u' u'l

= l' r 'llo 2 6l

= (6 - 6)a, + (-12)a,

= -l2ar+ 4a,

+4a

t.'7

unit vector = ffi =-ffi = -g'95a'+ 0'3 16a'

(b) Area = lBx Cl = "J12' + 42 = 12'65

(a) If the two vectors are parallel, then A x B = 0'

CHAPTER 1. VECTOR ANALYSIS AND MAXWELL'S EQUATIONS IN INTEGRAL FORM

(b) If A and B are perpendicular to each other, then A 'B = 0.

A.B = t1 x (-1)l + (b x3)+ [c x (-8)] = Q

.'. 3b-8c=l or

l.l2 (a) A = x2 yN, + y2 za, + x2 znz

.'. A

:.A

, l+8cJ

x = pcosQ

y - psinf

.'. A, = p3 cos2 @sin p

A, = p2zsinz P

A, = p'zcos'q

Ao = A*cos@ + A sin@ - p'cos3 QsinQ + p2zsinj Q

AQ = -A,sin@ + A, cos@ - -p3 sint Qcos2 Q + pzzsinz QcosQ

A,- P2zcos'Q

Aour* Arar* A,a,

(p'"or'@sin@ + p2zsin3 O)"0+(o' sin'@cos2 Q + p2zsinz $cosQ)ao+ p'zcos' pa,

(b)x = rsin0cos@

y - rsin0sin@

z = rcosO

.'. A" = 13 sin3 g cos' P sin @

A, = 13 sin2 9cosOsin2 @

A, = 13 sin2 gcosgcos'@

A sinOcos@+ Arsin0sinP + A cos0

13 sin2 g(sin2 9cos3 Psinp + sinOcosgsin3 p + cos2 gcos'p)

1.13

1 1Al.l1

a0

ae

Oe

B=-a +3a^+2aru9

-ar

a^

a.I

=-& t2a^-32fvo

+5a^-a.rdA

AT= 0T-0A BT= 91-33

.'. AT=3a +5a +5a - bu *a )=3a +3a +4ar I z \ y zl r ) z

BT=3a +5a + Sa -(a +a )= 2a +4a +5ax y i \ r yl r t z

fl1

Oga0

= 13a

L = 2a', + a', -3a1,

a'^ =

A,, =

^;=.'. A - 2ar-a,-3a,

la. aa a"l

n*n=l-r ; -;l

['3zl

l0 CHAPTER 1. VECTOR ANALYSIS AND MAXWELL'S EQUATIONS IN INTEGRAL FORM

Unit vector:

AT 3a-+3a,.+4a-a - = :--- = --# = 0'5 l45a +'0.5 l45a -t- 0.686a-Ar lATl ^!3,

+32 +4, ' v z

BT 2a-+4a..+5a-A =----+=0.298a +0.596a +0.745a-Br lBTl ^12, + 42 + 5' x r z

abn, = T;; ry'ff'an_toooldLl ? t= ;T;rr-' *^'(o'st+su'

+0'5145a' +0'686a')

36tt= 0.409 x 10ea, + 0.409 x 10ea, + 0.545 x l0ea.

O,Eu" = T;;m'^,,= i

I : .(o.zesu,+0.596ar+0.745a,)

+tt;.-xl0-'x45= o.otl3!^10ea + o.rlgzxl0ea +0.149x10ea-

.'. E,o*t= En, * E"t = 0'4; x 10e a, + 0'52; 10ea, + o'ega * ton ^,

(t.o x to-'n)'=2.304x 10-8N1.15 lFl=

1.16

4o * -J-x lo-e x (t o-'o;'36n \

AP = 0P - 0A = (2.5 - l)a, * 2a, = 1.5a, + 2a,

BP = 0P - 0B = (2.5 - 4)a, + 2a, = -1.5a, + 2a,