,Liyrrs
+ p tLnL:
?D l,nq ,
ccc+b3: q
b'd 3D p lqr.t I <txtbg+ cz =d
"t nr ,t*
& t/srhrr
W**
(*,y, z)= (*o
L..- , )a+re
z")
t€P
f : fo dirr4ioncl G pr""^ott
.+ t\) v .(o_ ,b, c)
\)ector
6e
n-L
y,a-ffrr\UU,+,C
-t ,cL /"^"
ac . '7.0
3'3" tew
t.
& .f ^2"-*..
Vo
8rn,.,.,.* n. -v.,rber
A
ok ''" /,r*
L-7"
u
/n.
1,.^.1, t sr,fojtct it^p-. l6ti
7 --to
d,rr ed,or.,rl rrdr"Vef
Jb Urn^Ja* *t
i)
,a- ,t^rrttt Agnw*rt
.--v--
t-: (r- t)t" tr,
+ vol.qre oe t(\
t--o ,/*\
r : fo 4c
t. \ f -_ fr
Sk-u,r \lrnus, .lo ^ot lrrtor:eat o,"A. axe al:o r..ot ^
t% \ie o0".. &flexelt p\* fdrd\tL\
f;.4 t le vctor a,,..d
?[0,-r,r), e(+, t:\*tt elua'hon s' tl'c fi|o atgr^e.,^t tt^at;"ins P t' Q
E= (t, q,-f) T(t)=(6, r, _3) t t({. g _j!\
p*,..,,&$\r .. (r(t),1tr),t1q)
= (+*,"+"_1 ,l -$)
, ?tA,n(s
S n t'* rn s!cr'c{ ': &ier""'va-d"u1 a 1or''\ o'-''\ &i'rectp\--..^-f
at A p\o,ru- rv'. tp"r.e ls &lun .qA
{, ". @ o^} o '"recL<
YffitH'l to ph'^t
^*fu"=,
^
(*,1, c> (,. ;r
" 1.,--r", -l'
"o"r"^)
ri" "1.,t)
ledov egn ol ,olor"q
-+ (},O'e>. (ol-.. ,t-J. ,7-a")=o
G(,-x") o@tf :J) - O(z--b.\ -o -.
t:*r.'{t^
-+ f r'-^i
@* t%. (* *A--c^
(o,b, c) = r^rorrnc\
Lo ? Ia'nas M'u'S[ -----*
;tiffil,
.o',,\,,r\Q
1^",*'J1o]filnue
(0,,.. top otoucNoti.u)
p*".,,r.or)
o 6rqq [Es b el.
*:.
wl{"e,v'. t p\o,.aq
"1, l td^c
bt,twetr
fl**
./
,T lelrseC-t rS
two nor.-.o-\ rt tcto( S
rh4
UI _:--)
\l ,- 1-
qcb :u
Aob \
\
\A\IT-\ \
)
$po'Snd'tw["r
a^ P,*J. ttrq "** \ be,tweBr" Ltq g\o.r..e5 :
1
rL +\-?", -3q*02=5
\% -*.,=''(r,r,-\)
"-.A -?a
t-* !-
6,-j, u)
n-
cos u --
-s A/ -\ l--J-
\ \fa " {21. s^ -_ cos [ {Biit
, $,,.r:tq^cz , t^t:ff.:=t:"i:.n' ['', , Y' ,") "'^{ !.w p\ru'e
"
L"B
Ii----P-
loo' 'b5'+cz''A\l
, of
;15.t,^, /,ntu [Y*io^s
f,;^n*' """ffa,.t4Q__, fo*, * ko,
l*r,T
ffu ,J..wr* ,1ou
){.
ho rt a Ft."J"^_"
nioclurt .-,.
oy o f*,..tio,-.,U o{ H^E + =i
e^ ra t
"*f;;::^., 1'"-
'nc\' {xfono-".* xt*, non^l,w
+ Sallq o ,.,r3qz=6
'.
't-0 7, :O t l^{r.. xz=3 a^rJ n^o*r1 ,..aore eolulions
r. we ho,re o' Yl^r# 4)
Jo 7 t : L -ix. "
,.r Vvxt vo,iiabte za-rs .,f,e2-
"' LuL x- , - L (. | ro{P
y"*^*ter) l torce o'\
on L> r
ux,{^.,{
r) -
tl,...e-r,^..- -(, -_
Fcrr'c'n"4dgf
ru
- a_st
- t 7
{1 -t I P*.*"ekic eg*olion5
ol^J \irv*
\n
lte,l\\S aQu(q,t,or
"La+ji-v _b
1C-u +-7 =+ OE ?i1 +;5a:I
d i 1
*-*a+\?,i | -12
"<_ a
_ k)
-t
_ rl
T; G" *"l
(o^\ For e*z lrtt vrs\
o n*t',1)^,*.*$r,@ O T*prsrrrLS MhN$
G Icontrsteiq)
\-. rc,
( tr\
Sct,ur,TIc,rtr\
crtsigber"t)
:o\".
^ll. l'^ts
, flJ,,cs ''^s'& br"drptf
q,3Tlowuicae
. cr,J
A-_
f "rl = l..-",r
I
o.'
d,z
€,wy {"
lar,
l.- I
oul5 rlQ
+ L*., "arrj
{6, f o .r,
cc lur-'
{ta\
COT T FTCTENT r;cr3q17--Z
a-ruY?- \ A[G ME Nrl-Eb
{-r-r"- -z ':
'J6 1*'*,
[._ B I .-l
l''-'l 5l
L-' "f ' l -1
lo
=__J
I
Tj":;Ti:* ilt Hi h:-*
'-.,
; ?" :, l;?:,"TXff*
,,lvd o+ [...t J* too
Abhtr'o-tio.. : T.-
,, t:,\ l 6!rJ.a.t 6{
i1l
:"i*'"il it;l _^- \,^li.t,,ngl6=.
6 A c t- SrB S-\ I-TwTTo\)
G,uet ' r.)q-?=\ ( r-l
J*utt?=
-z-=J \lL) Tf* t,,*Uulr 6.1
B^.ksulcs1 (u.to -) ( rlc1.- ] - I
-q , cc
3+ "
q+ L'(z) =( ;r'+3tYr)-z --l
g = -t, cc=!l
tt*,Lp $l.,,tio" :.
('t, - q,.)
t\ ?=l
l:l ,./+ 7
--'L Lst
lc r t _>
Loo o
";ct11+?^-S ) =Z-t
t1
a,^.c\
9= r*'t
Solution set , (*, !, z) =(t,z,o)* t(,,-r,r)
+ p tLnL:
?D l,nq ,
ccc+b3: q
b'd 3D p lqr.t I <txtbg+ cz =d
"t nr ,t*
& t/srhrr
W**
(*,y, z)= (*o
L..- , )a+re
z")
t€P
f : fo dirr4ioncl G pr""^ott
.+ t\) v .(o_ ,b, c)
\)ector
6e
n-L
y,a-ffrr\UU,+,C
-t ,cL /"^"
ac . '7.0
3'3" tew
t.
& .f ^2"-*..
Vo
8rn,.,.,.* n. -v.,rber
A
ok ''" /,r*
L-7"
u
/n.
1,.^.1, t sr,fojtct it^p-. l6ti
7 --to
d,rr ed,or.,rl rrdr"Vef
Jb Urn^Ja* *t
i)
,a- ,t^rrttt Agnw*rt
.--v--
t-: (r- t)t" tr,
+ vol.qre oe t(\
t--o ,/*\
r : fo 4c
t. \ f -_ fr
Sk-u,r \lrnus, .lo ^ot lrrtor:eat o,"A. axe al:o r..ot ^
t% \ie o0".. &flexelt p\* fdrd\tL\
f;.4 t le vctor a,,..d
?[0,-r,r), e(+, t:\*tt elua'hon s' tl'c fi|o atgr^e.,^t tt^at;"ins P t' Q
E= (t, q,-f) T(t)=(6, r, _3) t t({. g _j!\
p*,..,,&$\r .. (r(t),1tr),t1q)
= (+*,"+"_1 ,l -$)
, ?tA,n(s
S n t'* rn s!cr'c{ ': &ier""'va-d"u1 a 1or''\ o'-''\ &i'rectp\--..^-f
at A p\o,ru- rv'. tp"r.e ls &lun .qA
{, ". @ o^} o '"recL<
YffitH'l to ph'^t
^*fu"=,
^
(*,1, c> (,. ;r
" 1.,--r", -l'
"o"r"^)
ri" "1.,t)
ledov egn ol ,olor"q
-+ (},O'e>. (ol-.. ,t-J. ,7-a")=o
G(,-x") o@tf :J) - O(z--b.\ -o -.
t:*r.'{t^
-+ f r'-^i
@* t%. (* *A--c^
(o,b, c) = r^rorrnc\
Lo ? Ia'nas M'u'S[ -----*
;tiffil,
.o',,\,,r\Q
1^",*'J1o]filnue
(0,,.. top otoucNoti.u)
p*".,,r.or)
o 6rqq [Es b el.
*:.
wl{"e,v'. t p\o,.aq
"1, l td^c
bt,twetr
fl**
./
,T lelrseC-t rS
two nor.-.o-\ rt tcto( S
rh4
UI _:--)
\l ,- 1-
qcb :u
Aob \
\
\A\IT-\ \
)
$po'Snd'tw["r
a^ P,*J. ttrq "** \ be,tweBr" Ltq g\o.r..e5 :
1
rL +\-?", -3q*02=5
\% -*.,=''(r,r,-\)
"-.A -?a
t-* !-
6,-j, u)
n-
cos u --
-s A/ -\ l--J-
\ \fa " {21. s^ -_ cos [ {Biit
, $,,.r:tq^cz , t^t:ff.:=t:"i:.n' ['', , Y' ,") "'^{ !.w p\ru'e
"
L"B
Ii----P-
loo' 'b5'+cz''A\l
, of
;15.t,^, /,ntu [Y*io^s
f,;^n*' """ffa,.t4Q__, fo*, * ko,
l*r,T
ffu ,J..wr* ,1ou
){.
ho rt a Ft."J"^_"
nioclurt .-,.
oy o f*,..tio,-.,U o{ H^E + =i
e^ ra t
"*f;;::^., 1'"-
'nc\' {xfono-".* xt*, non^l,w
+ Sallq o ,.,r3qz=6
'.
't-0 7, :O t l^{r.. xz=3 a^rJ n^o*r1 ,..aore eolulions
r. we ho,re o' Yl^r# 4)
Jo 7 t : L -ix. "
,.r Vvxt vo,iiabte za-rs .,f,e2-
"' LuL x- , - L (. | ro{P
y"*^*ter) l torce o'\
on L> r
ux,{^.,{
r) -
tl,...e-r,^..- -(, -_
Fcrr'c'n"4dgf
ru
- a_st
- t 7
{1 -t I P*.*"ekic eg*olion5
ol^J \irv*
\n
lte,l\\S aQu(q,t,or
"La+ji-v _b
1C-u +-7 =+ OE ?i1 +;5a:I
d i 1
*-*a+\?,i | -12
"<_ a
_ k)
-t
_ rl
T; G" *"l
(o^\ For e*z lrtt vrs\
o n*t',1)^,*.*$r,@ O T*prsrrrLS MhN$
G Icontrsteiq)
\-. rc,
( tr\
Sct,ur,TIc,rtr\
crtsigber"t)
:o\".
^ll. l'^ts
, flJ,,cs ''^s'& br"drptf
q,3Tlowuicae
. cr,J
A-_
f "rl = l..-",r
I
o.'
d,z
€,wy {"
lar,
l.- I
oul5 rlQ
+ L*., "arrj
{6, f o .r,
cc lur-'
{ta\
COT T FTCTENT r;cr3q17--Z
a-ruY?- \ A[G ME Nrl-Eb
{-r-r"- -z ':
'J6 1*'*,
[._ B I .-l
l''-'l 5l
L-' "f ' l -1
lo
=__J
I
Tj":;Ti:* ilt Hi h:-*
'-.,
; ?" :, l;?:,"TXff*
,,lvd o+ [...t J* too
Abhtr'o-tio.. : T.-
,, t:,\ l 6!rJ.a.t 6{
i1l
:"i*'"il it;l _^- \,^li.t,,ngl6=.
6 A c t- SrB S-\ I-TwTTo\)
G,uet ' r.)q-?=\ ( r-l
J*utt?=
-z-=J \lL) Tf* t,,*Uulr 6.1
B^.ksulcs1 (u.to -) ( rlc1.- ] - I
-q , cc
3+ "
q+ L'(z) =( ;r'+3tYr)-z --l
g = -t, cc=!l
tt*,Lp $l.,,tio" :.
('t, - q,.)
t\ ?=l
l:l ,./+ 7
--'L Lst
lc r t _>
Loo o
";ct11+?^-S ) =Z-t
t1
a,^.c\
9= r*'t
Solution set , (*, !, z) =(t,z,o)* t(,,-r,r)
