CHAPTER 1. VECTOR ANALYSIS AND MAXWELL'S EQUATIONS IN INTEGRAL FORM
and OP.OQ = loPlloQlcos?
.'.
cos9 = cosdcosc[, + cosBcospr + cosTcosTr
t.2
(a) B=QP oP = QP+OQ
.'.8 = QP =
OP-OQ
|
\ t^
^ + ^za")-\zax = [-/a] = -2^,- 3a, -a.
\
+41, +Jaz/
(b) Assume the smaller angle between A and B is 9. The magnirude of projection of B on is lBlcos0. ... A B = lAllBlcoso
A'B ^ = --:--.'. 'h, |II|COSU tAl
(a, +za.
-:a.X-za, -:a" -a.) .,!I'+2' -t(-3)r
=
t'^l
t.(-2) +2.(-3) 114
-5
= -414 (c)
cosg=ry=
"lt-zl'
=
-t.336
-1.336 +(-3)'z+(-1)2
.'. 0 = 110.9"
-
-
+ (-3)(-1)
(d)
la' a.. a,'l
' 2
|
I
AxB = lt 3l lr l-2 -3 -ll = (-2 -9)a,+(6+ 1)a, + (-3 + 4)a, = -1la" +7ar+a, lA x Bl =
t
.'. unit vector = ,A
Jl l' B,
+7\ l' = 13.017 r
= -0.841a + 0.535a) + 0.0765a
lAxBl
1.3 Atx= l,y=).,7={ .4.=ar_lz *4a +12a A
unit vector
IAI l^-l
L4 Atx=2,y=3
ar,yz+4a +l2a
r._.------i "ll" +4'+12' = 0.0788arlz+03152a +0.9457a
A = 3aryz+2a +3a
B=4a +4a (a) A. B =
(b)
(3x 4) + (2x 4)+
(3
x0) = 20
AB ^=:-:---: =
= 0.7538
COSU
lAllBl
.'.
I
= 41.08"
(c) The projection of B along the direction of A is l8lcos0 lBlcosg = ^[4' + 1.5
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
A'C
(su, * za, + :a.)'
=
15+29+3
+ A'B
=
(r a
"
+ cra, + ^
")
=0
cr= -9
(sa,+zrr+:a.)'(4., *
Zar+ B,a,)
58,+418,=O (4u, * 2rr+ B,a,)'(r., -
B.C
9a, + a.)
34-18*8,=Q
+ B, = B' = 1.6
(a)
BxC
=
l4'5 -25'5
u'
u'l
l'lo 2r 'l
lu'
6l
= =
unit vector =
(b) Area = lBx t.'7
-
6)a, + (-12)a, +4a
-l2ar+
4a,
ffi =-ffi
Cl = "J12' + 42
(a) If the two vectors
(6
= -g'95a'+ 0'3 16a'
= 12'65
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
A.B = t1 x (-1)l .'.
l.l2
+ (b
3b-8c=l
x3)+
[c x
,
or
'B = 0. (-8)] = Q
l+8c J
(a) A = x2 yN, + y2 za, + x2 znz
x = y .'.
Ao = A*cos@ + AQ
= -A,sin@
A, = A, = A, = A
sin@
+ A, cos@
-
psinf
p3 cos2 @sin
p
p2zsinz P
p'zcos'q
-
QsinQ
+
p2zsinj
Qcos2 Q
+
pzzsinz QcosQ
p'cos3
-p3 sint
A,.'. A
pcosQ
Q
P2zcos'Q
Aour* Arar* A,a, (p'"or'@sin@
+ p2zsin3 O)"0+(o'
(b)
x = y z = .'. A" = A, = A, = :.A
sin'@cos2
Q
+ p2zsinz $cosQ)ao+ p'zcos' pa,
rsin0cos@
rsin0sin@ rcosO
13
sin3 g cos' P sin
@
13
sin2 9cosOsin2
@
13
sin2 gcosgcos'@
A cos0
A
sinOcos@+ Arsin0sinP +
13
sin2 g(sin2 9cos3 Psinp + sinOcosgsin3 p + cos2
gcos'p)
1.13
a0
fl1
Og
ae
a0
Oe
B=-aru9 +3a^+2a
L = 2a', + a', -3a1,
a'^
=
-ar
A,,
=
a^
a.I
^;= .'. A -
2ar-a,-3a, =-&fvot2a^-32
la. aa a"l
n*n=l-r
; -;l
= 13a rdA+5a^-a.
['3zl 1 1A l.l1
AT=
0T-0A
.'. AT=3ar +5aI +5az -
BT= 91-33
*a zl)=3ar +3a) +4a bu \ y
BT=3ax +5ay + Sai -(a +ayl)= 2ar +4at +5a \ r
z
z
l0
CHAPTER 1. VECTOR ANALYSIS AND MAXWELL'S EQUATIONS IN INTEGRAL FORM
Unit vector:
3a-+3a,.+4aAT --# = lATl ^!3, +32 +4,
a - = :---Ar
= 0'5
l45a
'
BT 2a-+4a..+5aA -Br=----+=0.298a + 42 + 5' lBTl
+'0.5
l45av-t- 0.686a
x+0.596ar +0.745a
z
z
^12,
a bn, = T;; ry'ff'an _toooldLl
= = Eu"
?
;T;rr-' 36tt
t
*^'(o'st+su'
0.409 x 10ea, + 0.409 x 10ea, + 0.545 x l0ea.
= T;;m'^,, O,
=
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, 1.15 lFl=
(t.o x to-'n)'
4o
* -J-x 36n
+0'5145a' +0'686a')
+
0'52; 10ea, + o'ega * ton
=2.304x 10-8N
lo-e x (t\ o-'o;'
1.16
AP = 0P BP = 0P
-
-
0A = (2.5
0B = (2.5
-
- l)a, * 2a, = 1.5a, + 2a, 4)a, + 2a, = -1.5a, + 2a,
^,
