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k"4 *o zPlru"e- (
Y--S3Sm-
------ilov loec
&oot
-rLrtrrrsitq:d
-at
Loe &-ssq*
rf !L
- O,2S 1e s| -l 4 8'asqX rdlz-
= D'o6z<
n
c-
8.esg * r6lL
-)c-
-1
.os8
-3 xro' -q 1.068*to = -12-It)
v/m
r.os8 fi t/-
C=
rii)
v
4 +
dJ: lrl.l< 'T
o' /o-u
y2 sio e do
o= =
lx
df
d:
{ltuo )
d7,
C,'.;
e do
d{1 d7
-.-
-
o( (
4
(Jls,;saed{ d7='l G.e =( JJ 4
o
oo
9rr
=+ .tJo [ft.'o]*d$ o
9n
i
c-ol a{ JE^,. +
o
c-If
15 ^ ,
-
;)Gc-,1;Ddh o 9$ ,.II rg [ol" =r".& J( a& =1, -/ 4 t D
x2ra = "3 a
A = T\t3
q/ ,Y=O.2S ro
:
r,
( o-rs)
3
= O-o,s5as -3 x z.tl @ = O.o49oc t'i,
Y
'
Q= trv3
= ., 1o. as;
3
*x o.o+r8ls: , =
?
= o'rar+6 c.
@
Ezplacn lorrt
*,f4\.,abut
cXL"a;ia-t
co'old;'taf,-C.
,
'fhL
fl\rdr;c.!-
co-otd,inorh'
.'sytn'"
'ha-t
?
CD-srd;r,ch- Ct, {, =2. =2
Y ptona:-
A yb''az o[ xodi^* ' y il *nocrn A)
2
(R-
qb"-M
ptonz.
Q- 7lon"n,
A
vertca-Q
ptcr'* 9
g&'*td
la x*Y Pttn.a oL%L+ Y trilh 9@^W *. x - oli'r'
)
z.-
: Vlant- '
fL )' a -hoazonto! Pt"cnltrticx b garatt-a! fo L-Y P\a"uo '
2 -A*wt P\rn.l-
;t,"? o prnb p
-in C1tndacal
Stptr" :-
-'l-tv- 3
1* r
vot-ug
plonc-
I -n_ t\)
Zyl"*'
rLZ cLJ-./
o
ZYzE
.4.aft
-@
/
I
x
'fhr-
ptn"u
,rbwrt; ,l
1,r s+or$
"tQ
con
'be Pou\t P fayx,e-t zntad a-t f r, $,2').
)J^t
Vec-l3r;.
a,
-Thsz- au-
qg
K,<
./,
\I
thtt
Vecfetr at , a6
drl
UnlL
trrz' n'' 'be
o
luvrEd/ u
Unit
Co'e
tlontr q
_.1
) ()',t,
lr) co'otdrna ta
on,h
"4
v
vsc-D6'
a?
:-
Tt
D2^
)r, o- p\o,na p"x.ottd .b x-y
tt );r- ferpon&euta.t *" 4t ahe* q,i ,-
veds.
rt
rt
i-r d,to pa:.allel -b x-y plone . poirrE )i ttru- Ahzrtrnn o$ )"c-""iX
4,
)
Unt vects+ E It fu. ootailil lo =-ait.
('
I
peitii"
Vecta.e:-
'Th2-
posiG-oo
d-Do,^i | - lY Prj.6o.ra"tirl
eu* eoV.
ofi
vecfo+
q16'd,lc"l' co-q&'nofo
+ ?qiip + ?=
j^
42.
otrUdrrrrg. Co-edinqta, sd^b')
a-*d?
[*
-a{
\
4-
Canudu Co-otd-.irato-
\
--
tt '
t"\r,
a-
po'
Q,
z.)
tn .t fl(;rlnc"!
uft*'
Gc''p-a"n 24ch c-.,- otdr'oo-b- vot, a 3x dr, dz- X vA$, dJ{Pun1rol )o-Z*
3G- e io nol ,o
A;Wi.t
lo,2+n
,alua;nhb
ji,n,
c'
th-e.
..,; 7rv- a^L b Utn or'd4. 4
ltl,
a
V?-ctbe )9-s.1" u(
dtlt",r"t".!
fi
A)veYt' A-t'.
d"d+raoo(+A'&'
l.u& of di ia,
Moant
d"
=
+t
----.....=.=-
laf [ . .i oln = +"( aqz+dzz. Votum-e- : -
$r{-[arznfLa-l
'rht-
st jhz lLoA*
cl-.'t#rantrt-l vstumz-
-fltnen 6Y,
dv . Ar' rd$. dz Av = Ydv d4 d=.
Sl*,,r"t.1 -stdrnobo
Arrlacz t-
Ad/\ttn
,Lt'nrnL
"l-
rr,t'n&;M.!. 0
-
.t
i\affi," LNI
+
d9r
l'6 L dcQ
th'/' tulp""r;"l Aut[ac-o-
-d"* )", t
R,
= rd4 dr-
I -d,
di
vecte+
n*.n&
1"
fl,-
dr[P"ltuaL'
ds4
=
'Thr- h5*,""t;.] -Plart- n, dsz8.
^b*
'.tar}:o<.
-ri)
&a
=
,4+r+l
vllt-
""r-,(+l 3 \ c."-l/ 3'1\ I )
9-_ 36.66 lit)
+=
+a;'(31*)
t-;' (y")
, 4-
a6.<6
q-7t",-
a,' C
,aLl:.,lact vec[e<-
notrna-l
,h z-
vAQ&,7-.
{. h VaL-r-e- a=p. !--t, rb y = ,I;{z;;:
=
h
a"a-=4.
'Taat"^l'n.**,h. ea*hian b
=1t
neu'a,t
"fr;U^rrl
& z-= I
c--olit
.o.te-
i
?
fuud',
alrvLi,rt,-on of ALNIA
.F.,,o
ti) ?o-tnL cloxy:-
lr[
,-
&
4[ L1 ), .t )D,od )t)'-* :.hA^Ct- q jot,rt c)1
B
1l'o
po-
dJ^*"A
Y'ad-.otlld
0"-[i""..&
alonX rolu. o+.
G";dr'"
Tn
d
ditr)f&,*,& "au$ore- ds' cr'-e<&inqt;.'
spl.etiea)-
ds- v'
sin e de dQ
-- eL si,) o do l^1.K.
d
r
tr*= --t cls =
cQ,,u- ii> -.->
D:
Js
cr6
oz s.l o do
ooint ro -Q 4o
, 4o o-'
.,
dQ o'r
c.har9a-
v'
)tb^, l,
i2,
---!
a?
ct,"
lol | "lsl ces O '
d{
gtt?&
Y= O'
-*sueclodq'coro'
a
+a* p,.t -D". a3
g,
O
g dod+ g.; 9=_-
4rr
*o
..
= -g-- ei* e Ae d0 4n
J,;laL
\k*
ry - jd' er
Ls
7"'"'
a-i
tX
Q s-o de d{
JS0:o
$.o
+ l-_*"]l [o]"'
= 4lt
L
=
*t_.., " ",-,{ LanJ
-_
a [ _(_,t+)Len) lfL
-- Qz ?
At- = & th,, ut- "Jt * ;'
*La
-1r.
F"^,
quo!
**y
A" A'2 ch*tr
Gnd*
o,r
d^* **Y -* l"t -Pfu xaA)u.t, [' 0rr,'-,<-,f"J*ica'Q- t':i9"
lW u'-1..* i' tr" Xh- B it xo&ail,.tr *in
O,rhrror& oJo'6
a?
'
a rL
1tu- )tux,
h= 9r m
v
:, Q,$ts u? L"v'*l x Xhz eha-X* * b"' 1(i"d,c.l i.t
.aeqrorr,"t
A )rt
&,68 J'
fu^'Le'
a_i +98'aB
t*
sldr-
":{"*: +$r al bo+t"t"'
B =] & fi & = "d{d' dz Aq r.e.tr=) ,.8' d? " 'Dtt nL b* cXu-ariraL
[^1.K.:r-,
th' W r' 'beHa*
en""t *an wo
,
(omToncnb
ivt z'a'ti'g
.'.D'= $tsu< J'i\'o
))
=l
L
Z=o
ev
Io "rd{
fu--o
dz'
'
:
Y.Dy
[. - "]
L, "
-"]
t. - YD7 L&r
a anr L
}.=
h."'r; Dr
D'
-.> qv
d7
=4
SnrL
P"t Ql,
= ?-
tl*'
=fu^7 rrY a
-_->
-'D
Eo
c-
{-
a.t
&n€or to'
A \*6
t?frro,h,"dr.r--
-hu
verrhrr_u a-L p,ta,o,o),
d.arrff* $ t-. d^L )ocul"A a,t, -enc.l, al ,Or!4 vpxb.i-e* Z{ ba an}r6rltd-ru b L"^ttd i^ mo'6fr1a' 4 #* * zactlht ** f.*, +,jS Cl,LaLf
' A oo7
.':'('i
Prr
to,o, &'/i)
)>
e" t *t , 'fz,o)
7, lz, o ,a Fero- Q, Q+
T=
Q, Qg
4lI -2
R+=
6o
aiq
lilel'
Y+ -ft ") o$ + e{7az + t(Z a2 -&&" =
- [oai. +
l*il
4
"{; +8- .={1; RIL
l6il - Aa'i * ,Ja {
.F
-Lrd
\t'
&,0+ ^L 4n s, (d\;)
(rx,ot)
-)
Az-
(.
(txr;)
4nx8.8qq r]!axtz
aai
+ ata
;21
\nQ
.'/-a a; + ac-a7\
t.
fla
/
=
lxlo-
z {622. 1s6 x tJl
(-aof
+ar6'
= &.t6ax rJQx,oL ( -afi + a,laaz) .- i t6,3x (.aq7 -t ec oZ) = -+2a.6 -
Q.
z-
-t
6rl . q.qb dZ.
":1;::..'...
-.,
l-otce
Qe @q
f = Qr
Q,+
4n€6 R; z *-+ -> --+ Rp* = Yt1 - Yz-
-
)nl6 dZ
--
o.:A
\naq\=
-G"Z +/Z-a?)
- JT€
+pGdZ
fr+a+B =,fi;
.--.>
Ai
a-'a+
n2\
=
,a,-n + arf; e
AZ
,fG
.'.
F'
.=
(r*
r{ -Jao.{'+ 5+ a.,{a aZ td3)(rxt;3) l'ad I
4 n x 8. 8sq Y r6l'xtz- L
ar6.3(d;
att'a
-Jtif
{G+ a.t;aZ)
d *3.,+.4Bsq
+6rr.+qbqZ.
Fuca- Q3@9
? = QsQq an
---> .Raq
et I p]* 1r
;r _€ a{;qz -
ae,, -
(-fi -{2"{)
ai *'6of + a S; ciZ
l*;l
= {r.+ 3+ 8-
--> &R39
=
_
R"a,
=
= tl.l2a.L
"fa4+r'tl.aZ
"i\ (tx,o-3)Ctx,;E) \
4x
a rg
x
(fi
,fGI a{- + 6E,i! + I
s.asgx rJ"xrr-
L
e{i _J .j
V t"-
{ " afn fi) ars., di + 31+.+8c ( +6rr.4qbdZ. aru.:
Acc-drnX
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to
o uz
S*ga-rporifio^ G|" Pai"",:Pt'z-,
-- Fr *Fp t
{*.
F3
=='++6-18 -- | 8 39' sas aZ T.
fuinu
.b^-Q,
zLe&it $aa d'* lo i^[ii'h' )i"'-
Cl^a)tr. Gnoide,t
,t-uh
a in|;,t4, )iv-
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d'at-yz- ubaw
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C'
B
r
a?
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ch.anoo-
);,,2'
6
.on
g{1 v'aqb a'L 'o 9o'''r .P.u
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dula"co-
!. A-
.
.Le]|'.-"o"
d-c
d.c. e
= 'rP --
)bt
an
Yd-Q
[';i - ="2)
Unit Yectcr is, . ----, l-. C\R = t<
lR'l
=
Cn4
r
z'adtt '
dz
=
Aij:tancp-
R
Ox)A.
flue* a-t
clJitr{-a'"dJ A-a,net'l'
d.q = R"
b
2-
e. d-l.
otq =
-Th-a-
,lt-u
' -r^
{;v*
o"3 r, ,t :. \P = \ay --' rf,j--}, = 'zaz-
thz -duhic +ila ef -ba d,lleuA;of -dutpnt clp de da'=
4n €oRz (^,
ft d0
4neo(Jy4t;)z L 'z - Componen b , ha
dE
?-d-
8Lr"-.f,"q
)1@,
,$e.caar,
4fl
iEyat.w,
'B5
czrJ.c,,La,Gd
a) ,
.'. z-: r d= .,
eo
(JYz+z,- )
I 4ne"6rz*;lrL
J -&
tan
o.
o.o
6r = tM-' (_ ^) __ _tr/r
=
alooue-
cD
--1 O= tcln
n-/,
. L aoJ
l
Ll uahoa
:g {r+;:
bl.e. -\ Jo..al -dufu;"
(?-
-z --co 'l^t.ort O :
|-6r 'z-
2
--'iZ
$Aa
o,o
=
Fot
^i
"rdz-
q
)-E
rl \rxt'
?.-
J 4" r" (r""*to,&o)3le
a^
<€c^O d.O a,I C 'l
(y24rLtor'o)31-
_tr la
,al
1,lo Pr
Y
J/>4ne"
-rr
?-f\
>gecz
YB
d"€"
G
s".3g
stz-
=rr(
t
+Vo,r's)'t" 6t \ cl c-
gz se-c'o ) '- '{ 3 secz o
=
aj
4;;. J "'sece _trle
Pr
I c-," anai
4n€or J
=
-lv- [Lr-;1"1"n 4nEoY )g"-
-trle
=?-
4n€ot
=
F''"'
o
tt-
4ng-,r
2n
-
sin
(-rrl,)]
[,- (-')]qf
-\
< a-2 fu
2. aq
4n€oa
€o"Y
-elzct'ic -1142 a[lc'e Eao'u'uL 4 fure tot*L ef iniini.fo, );"o ct"n'ty' J.
+.ld I.
zu-o, i) o-vsob<' ctra,l o[ o[ dwexXt'u-'a prove- )hab uav / fuNE 3 oo6 Uu;X da
-
Aoln.r
Gn a
dtt a
Vuc1aa A
cu-lolA:VxA a
Divetync-o
= e. (e YA)
rl
ciiii ti l;
ii Il
ii
i
ii
ll u
ii
Qi"axVnao-
ol a Vedrer, )*
,-
ir
ii
i
X { -*[o-
,
!t
i il
fi i t
lz
v'
(w<
aldv = J *^s J
_->@
hrtp- s-- ctprecl a.u fiaco endrr:inX )ilo .ido <1" @ -@ u* 3"t,
fi
IiJ
-------> O
exaLs-te
J Vol
il
:t
quonl'-t
)^rff"l "t l*U atd-at .evet o- Vo{.mo IVol v' (vxA) dv =J""av --'2@ "'Ar"*7** +ur*n.,'o' -ieet1;; f-nt
ii
fi
aealo-t-
.'. V. V)( A = ao
lr
$
o-
l^l
6n''''toLum-e-
= J "'dv Jo"oas I vot
' ,L V
L^ t ,,h,ko'o
tyodut '
^ ' " --6.,p-e..or*
t+'
zelD .a!t
cL''"J a a??fu "'trird *" 'b^L
LDu
.
t *dv = o , 'h''n's'
oo=o
Vot
&usur,t" thi-o r; 4n O v. (wxa) = 6.
U** it u
Ptn*zd'
il G i,,ar. A= (y ces ar )ai, + (y+a{) a-=
.
n-"1
v, a
CNp:f2qsJoo2 (6) io, jt* M?oJ o2 o'2 I lu* 63 d'2:i Wxe= [4- tJ Wxn=[4'xlu* ai ^ztat a/a=l= a/aj lu,r, lu,uil
I A:c AJ
n=
I
",u, lY*"."" o
II
|
Y+e*l
I
il it it tl !I
'"1&
i1
ii ii
* o=i-^I y,- o*'l
1i
!l ti
i!
LdJ ) a*[o] * os[*-,"r-(v-"-l *o'(o)
ll
vxn = -&5 4.
rr"tl * "r[- tv'*''-*t'n""] '
'r[
(v+e")
)^ t-roo vecl'a: *o c'a'n'*o'\
e1U^d'tr.
L
to'gd)nqba
*qz ) B= -Qr- * Ttt ' Aaz" y>lanz' W to uztpe",d,u.-la* ver-f,r' un-ub cr. Gmp,-.to Novll oec'*7(8) B '^ r^r''r^^ A^ Lo o Conta-inln2
a* A=
Aa.r-+
rta5
Bo[,rt-o :'
b4
of *t:ot? ii -o- ,,omenLal on *o*ro. ca }cls- eb+atn
a*o- elds and gt&Ly^d".
.bnz netmag' ,,-l a&o-tL/
x"*"1Ltr
3o d3=r'sao
dQdo' = R2 s&tO dod$
{ v-7;Z ofn^ Areo = e= ( [sa,o do df.
As
))
oo
d-
-)D
= R, \ lD
d-
*nl I '+ =;tJi'-"-c-"1Jt
-.ii-
d*@)
= -*'J [-,
-,ldQ '= n2J a aQ
oo il
^ = R" Lue]
o/-
= Rz faa -o)
J
ii
= A..R'
i,
,,ii
pxova ,Div u = f 1 &ln:'1\e- dpi^ic
lli" llllli*
u*r,,a
)a, . @rr'tlmev a'or) d'*A strtlaco 'ti'u'
G.,,.r*,
at "t ', erLu,n to -il4t' t,iu,l. ct^ary-.n-A-"A AZ* fh} i2 knst>t a-a .Qot o lau:.
4t* i-Z
Yaltzmc:lbo!. y.*tzoenLat;"
lii
iL2
&vet-q-nn
otr
0
lri
d:v.
s= -lri-
.
lij lii
Fr-'n
Av->6
d P."1"
A"
:?_e?!+E>y'Dp= j-r_ Tt ' aqa_,, 12
)ari,
-E6v",A"
)*
-6vro
6 >.ds
S-?-CJ o =L-'' b\/ aV+O Av
Scnt,,p,., O _ tv Av+o
'
G-u'o
Av
dtv.D-
)o'o,
-elcclu: "'Jt"* d'"si[
Lf = Q =3[ods
lir
l; li l.
t'i"'a"
o[
?.
'*t{aa'
'.:t,
l^
lgataA aL i,rp,t"h -lr% d.f€,,- fitu chaqf )2 ir P- gs nc-f m , -P"! #'d t A,A7 e
.f
\'-g,',-==, ci>
0r.tg,i"
.9ol"lLo
P[0 , 6,
tiu
r)
ond
. d 6',5, ,) Nov f/Duc dmr
*
.-
$,u,s) (o,o.o x P,e ).
5
' ?y,
'
36 nc lro
,
?=
P"
u -a 2=e at dnotyzbt9
Soln
tj,
:
3n€o'r
i7 v/d
? aL
".i8i" .su toncg r= {(a-o1za ( S-o)" ={ 9}zs
E:
=J
3f
m
3on = 92. +s vl'a 3 n x s. ts9 *r? "x ,1E9
(D? o-t. P(o, 5, pu
ton co- ,
r =',/Q-ffl!-4-= fiTi[-r-',/IF=s-
..->
E=
0i,
30
^-
an^ 8-8sg*ri'xS
= \o-1.85. v 1.,
? otp(s, 6,r): Ftr
ton
r= ,t 1.-6r.l \ =
r,z
9:Yr)
3on A rTx 8.
EsqXr!r2*S
=
to1.65- Vl.n