06_Plano_UFMG

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6
 !" 
1
 !"#$
 !"#$%
%&'(
 !"#$#%&'" ' (#fi%*+,- (# #./'+0#$
(# !1'%-$ %- #$!'+- # $/'$ !"-!"*#2
('(#$ 3#-45&"*6'$7
$)*&'!"$+
8(#%&*fi6'" ' #./'+,- (- !1'%-
%'$ 9-"4'$ :#&-"*'1; !'"'45&"*6';
$*45&"*6'$ # "#(/<*('$7
=#6-%>#6#" '$ !"-!"*#('(#$ 3#-452
&"*6'$ (- !'"'1#1*$4- # !#"!#%(*6/2
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# "#&'$7
,#-.#&/0!+!'$+
?'@#" *(#%&*fi6'" ' #./'+,- (' "#&'
%'$ 9-"4'$ #4 ./# 9-"'4 '!"#$#%&'2
('$ %' '/1' '%&#"*-" # "#6-%>#6#" '$
!"-!"*#('(#$ 3#-45&"*6'$ (- !'"'1#2
1*$4-7
 !"#$%
 !" #$%&'()*+'
 !"# $% %&!% '%((%)%* +,-.fi0%12( 3&, 4 '2((5+,! ),fi6.- %
,3&%782 +,92-.%! )% -,9% &9.!.:%6)2 %',6%( &1 +,92- , &1
'2692 )2 '!%62; <!41 ).((2* %'-,6),12( % -,'-,(,69%- %
-,9% % '%-9.- )% (&% ,3&%782 '%-%149-.0%* ),fi6.- &1% -,9%
'2- )2.( '2692( , &1 (,=1,692 '%-%1,9-.:%)2; >%1?41 @2.
'2((5+,! 026A,0,- %( '-2'-.,)%),( )%( -,9%(;
 '-.60.'%! %((&692 % (,- ).(0&9.)2 6,(9% %&!% 4 2 '!%62* 3&,
(,-" ,(9&)%)2 (2?-, 2 ,('%72 9-.).1,6(.26%!; <!41 ).((2* .-,B
12( 026A,0,- %( (&%( ,3&%7C,( =,-%!* +,92-.%! , '%-%149-.0%*
?,1 0212 (&%( '-2'-.,)%),(;
 !, -.)/*+' 01&/2 (' 32/$'
D,E% A = (x1, y1, z1) &1 '2692 ',-9,60,69, % &1 '!%62 Π ,
~v = (a, b, c)* ~v 6= ~0* &1 +,92- 2-92=26%! %2 '!%62;
F212 ~v ⊥ Π* ~v 4 $%&'#" G2-92=26%!H % 92)2 +,92- -,'-,(,6B
9%)2 ,1 Π* ,6982 &1 '2692 P = (x, y, z) ∈ Π (,* , (21,69,
(,* 2 +,92-
−→
AP 4 2-92=26%! % ~v* .(92 4*
〈~v, P −A〉 = 0 GI;JH
94
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
6
 !" 
 ! 〈(a, b, c), (x− x1, y− y1, z− z1)〉 = 0⇒ a(x− x1) + b(y−
y1)+c(z−z1) = 0" ! #$%&# ax+by+cz−ax1−by1−cz1 = 0'
( ) −ax1−by1−cz1 = d %* &+,+%&+ &+ x" y ! z" -.+) /
ax+ by + cz + d = 0 01'23
4/.# 5 # +6!#7* 8+9#: & ,:#% Π'
 !"#$%&'() ;* 0#3 ( ) ~v = (a, b, c) 5 % 9)#: # Π" ∀λ ∈
R− {0}" λ~v 5 .#)-5) <+. 9 % 9)#: # ,:#% '
0-3 =+9>+-# 6!+ %# +6!#7* 01'23 / > +fi>$+%.+/ a, b, c /* 
> 9&+%#&#/ & <+. 9 % 9)#: # ,:#% ~v = (a, b, c)'
0>3 =#9# &+.+9)$%#9 , %. / & ,:#% " -#/.# 6!+ /+ #.9$-!#
<#: 9+/ ,#9# &!#/ &+ /!#/ <#9$@<+$/" &+$A#%& !)# &+:#/
:$<9+' = 9 +A+),: " % ,:#% &+ +6!#7* 8+9#: 2x−3y+
z − 1 = 0 .+) / 6!+ /+ x = 1 + y = 0" +%.* 
z = −2x+ 3y + 1 = −2(1) + 3(0) + 1 = −1,
, 9.#%. " , %. P = (1, 0,−1) ∈ Π'
B+C# #$%&# 6!+D
• /+ P = (x0, y0, z0) ∈ Π" .#: 6!+
ax0 + by0 + cz0 + d = 0⇒ d = −ax0 − by0 − cz0
"
&$E+) / 6!+ P 5 # #$(3!' & ,:#% Π'
• F+ ,:#% Π > %.5) , %. (0, 0, 0)" +%.* # +6!#7* 
8+9#: & ,:#% /+9@ &#&# , 9
ax+ by + cz = 0.
= $/ a(0) + b(0) + c(0) + d = 0⇒ d = 0'
95
 !"#$%
&'()*"% +,-,., !"# $ " %&!"'() *%+"# ," +%-" &!% ."//"
.%#) .)0-) P = (1, 1,−1) % -%1 2)1) 3%-)+ 0)+1"# ~u =
(2,−1, 3)4
5)1) " %&!"'() *%+"# ,) .#"0) $ ,"," .)+
ax+ by + cz + d = 0,
-%1)/ &!% 2x+ (−1)y+3z+ d = 06 .)7/ ~u $ ) 3%-)+ 0)+1"#8
9 "70,"6
2(1)− 1(1) + 3(−1) + d = 0⇒ d = 2,
.)+-"0-)6 " %&!"'() *%+"# ,) .#"0) &!% ."//" .)+ P = (1, 1,−1)
% -%1 3%-)+ 0)+1"# ~u $ 2x− y + 3z + 2 = 08
 !" #$%&'() *+,)-.&/ + #$%&'0+1 2&-&3
45,-.6&1 7) 2/&8)
:%;"1A = (x0, y0, z0) !1 .)0-) ,) .#"0)Π % ~u = (a1, b1, c1), ~v =
(a2, b2, c2) 3%-)+%/ .%+-%02%0-%/ " Π6 % ."+%#"#)/ %0-+% /78
<"+" -),) .)0-) P ∈ Π6 )/ 3%-)+%/ −→AP, ~u % ~v /() 2).#"0"+%/8
=1 .)0-) P = (x, y, z) ∈ Π /%6 % /)1%0-% /%6 %>7/-%1 h, t ∈
R6 -"# &!%
P −A = h~u+ t~v,
)!
P = A+ h~u+ t~v,
)! "70,"6
(x, y, z) = (x0, y0, z0) + h(a1, b1, c1) + t(a2, b2, c2), h, t ∈ R
?@8AB
96
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
6
 !" 
 !"#$%&' ()*+, - ./$0$1$ 1! !34)56# 1!"#$(), 1' 23$4'
Π ! '5 6!7'8!5 ~u ! ~v 5&' '5 6!7'8!5 198!7'8!5 1! Π*
 2$8798 1$ !"#$%&' )*+,: ';7!0'5


x = x0 + a1h+ a2t
y = y0 + b1h+ b2t
z = z0 + c1h+ c2t, h, t ∈ R
()*<,
 5 !"#$%=!5 ()*<, 5&' .'4/!.91$5 .'0' !34)57!% 8)$)'9:
"$(.)% 1' 23$4' Π: !0 "#! h ! t 5&' .'4/!.91$5 .'0' 8)$;:
'!"$#%*
<=!'8,# >?@?2? > 23$4' Π "#! 2$55$ 2!3' 2'47' P =
(1,−1, 1) ! - 2$8$3!3' $'5 6!7'8!5 ~u = (2, 1,−1) ! ~v = (2,−1, 0)
7!0 .'0' !"#$%&' 6!7'89$3
(x, y, z) = (1,−1, 1) + h(2, 1,−1) + t(2,−1, 0)
:
! 5#$ !"#$%&' 2$8$0-789.$ 5!8? 1$1$ 2'8


x = 1 + 2h+ 2t
y = −1 + 1h− t
z = 1− h+ (0)t, h, t ∈ R
<=!'8,# >?@?A ( !"#$%& '()&*+#, -( ". /#*#,(,&0
1*#.&2? @$1'5 '5 2'47'5 A: B ! C 4&' !0 394/$ 8!7$: '5
97
 !"#$%
 !"#$!%
−−→
AB !
−→
AC &!"!$'()*' # +*$*,!,#-$*'# ./0* !1/*23#
 !"#$(*, 4
P = A+ h(
−−→
AB) + t(
−→
AC)
#/
P = A+ h(B −A) + t(C −A) 5 .#' h, t ∈ [0, 1], 6789:
!' 1/! P $!+$!%!)"* /' +#)"# 1/*,1/!$ &!%%! +*$*,!,#-$*'#8
 !" #$%& $'()*$& +,-+,%./$/.&
&'fi$)*+% ,-./ 6 !"#$%& '( '%)& *$+!%&,- ;!0*' #%
+,*)#% Π1 ! Π25 .#' !"#$!% )#$'*(% ~v1 ! ~v25 $!%+!."( *<
'!)"!8 =>*'*<%! 0$12"% 3' 3%)4 5"#$%4 # '!)#$ ?)-/,#
@#$'*&# !)"$! # !"#$ )#$'*, * Π1 ! # !"#$ )#$'*, * Π28
;! θ @#$ !%%! ?)-/,#5 !)"3# "!'#%
cos θ =
|〈~v1, ~v2〉|
|~v1| |~v2| .#' 0 ≤ θ ≤
π
2
678A:
67'85"% ,-9-:- B!0* #% +,*)#%
Π1 : 2x+ y − z + 1 = 0 ! Π2 : x+ y − 2 = 0
;!)&# ~v1 = (2, 1,−1) ! ~v2 = (1, 1, 0) #% !"#$!% )#$'*(% *#%
+,*)#% Π1 ! Π25 $!%+!."( *'!)"!5 ! +!,* &!fi)(23# &*&* +#$
678A:5 "!'#% 1/!
cos θ =
|〈(2, 1,−1), (1, 1, 0)〉|
|(2, 1,−1) |(1, 1, 0)| =
|2 + 1 + 0|√
6
√
2
98
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
6
 !" 
 !"#$%#!& cos θ =
√
3
2
⇒ θ = arc cos
√
3
2
& ' $(()*& ($+'%,!
-.' 0 ≤ θ ≤ π
2
& !+#'*!( θ =
π
6
/
3!fi+(56# 7897 0 !"#$% &'(&#)*+,!"'&%-8 1!%(),'"'2
*!( ,!)( 34$%!(& Π1 ' Π2& ' ('5$* ~v1 ' ~v2 ('.( "'(3'6#)7!(
7'#!"'( %!"*$)(/
Π1 ⊥ Π2 ⇔ ~v1 ⊥ ~v2 ⇔ 〈~v1, ~v2〉 = 0 08/9:
:;!'<,# 78=898 ;$,!( !( 34$%!(
Π1 : 2x+ y − z + 1 = 0 ' Π3 : x+ 2z − 2 = 0,
7'")fi6$*!( -.' '4'( (=! 3'"3'%,)6.4$"'(& 3!)( 〈(2, 1,−1), (1, 0, 2)〉 =
2(1) + 1(0) + (−1)(2) = 2− 2 = 0/ >$( 5? !( 34$%!(
Π1 : 2x+ y − z + 1 = 0 ' Π2 : x+ y − 2 = 0,
6!*! 7'")fi6$*!( %! '@'*34! 08/A/B:& %=! (=! 3'"3'%,)6.4$2
"'(& 3!)( ! C%D.4! '%#"' '4'( E θ =
π
6
6= 0/
3!fi+(56# 789> 0 "'"!&!*%.$ & &'(&#)*+,!"'*%.$ &#/
0'& 1&0" & !"#$-8 F'5$ " .*$ "'#$ 6!* $ ,)"'G=! ,' ~u '
99
 !"#$%
 ! "#$%& Ω '&! ()*&+ %&+!$# ~n, )%*-& *)!&. / )
r ‖ Ω⇔ ~v ⊥ ~n⇔ 〈~v, ~n〉 = 0 01234
r ⊥ Ω⇔ ~v ‖ ~n⇔ ~v = λ~n, "$+$ $#5 ! λ ∈ R 01264
785 +$ 129:; r ‖ Ω
785 +$ 129<; r ⊥ Ω
 !"#$%&'() <: 0 !"# $%&"'(# !) *+#&%,* =)>$! r !$
+)*$ ) Π ! "#$%&,r ⊂ Π .);
• ?&8. "&%*&. A,B ∈ r ) *$!@A! A,B ∈ Π2
• 〈~v, ~n〉 = 0, )! / ) ~v A & ()*&+ ?8+)*&+ ?) r, ~n & ()*&+
%&+!$# $ Π ) & "&%*& A $+@8*+B+8&, .)%?& A ∈ r ∩Π2
&'()*"% +,-,., C$?&. $ +)*$ r ) & "#$%& Ω, ?)*)+!8%$+ &
($#&+ ?) m "$+$ / ) r ‖ Ω 0 ) "$+$ r ⊥ Ω4, $ "$+*8+ ?&.
.)5 8%*). ($#&+).
r :


x = −3 + t
y = −1 + 2t
z = 4t
) Π : mx− y − 2z − 3 = 0
 !"! #$$%& '()! *+( ~v = (1, 2, 4) , % '(-%" .#"(-%" .( r (
~n = (m,−1,−2) , % '(-%" /%"0!1 !% 21!/% Ω3 4$$#0& 2!"!
100
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
6
 !" 
 !" r ⊥ Ω⇒ ~v = λ~n
⇒ (1, 2, 4) = λ(m,−1−2)⇒


1 = mλ
2 = (−1)λ
4 = (−2)λ
⇒ λ = −2⇒ m = −1
2
#$%&'(&$) *'%' !" r ‖ Ω) +","-." &"% !" m = −1
2
/ 0$ 1'.$
+" r ⊥ Ω⇒ 〈~v, ~n〉 = 0) "(&2$)
〈(1, 2, 4), (m,−1,−2)〉 = m+2(−1)+4(−2) = 0⇔ m−10 = 0⇔ m = 10.
3$4$) r ⊥ Ω⇔ m = 10/
#$%$& '()*+,*-./ 0*()+* 123(/, * *()+* +*)3, *
123(/,4
 !"#$ %& '(#)%& )*% '#+#(!(%& Π ! Ω, - .)/!+&!0*% !)/+!
1%.& '(#)%& )*% '#+#(!(%& 2 3$# +!/# r 43"# !53#0*% 1!&!"#6
&! 1!/!+$.)#+, 7#+# /#)/%8 4%$% r !&/9 4%)/.1# !$ Π∩Ω8 #&
4%%+1!)#1#& 1! 53#(53!+ '%)/% (x, y, z) ∈ r 1!:!$ &#/.&;#<!+
#& !53#0=!& 1! Π ! Ω
34!'5,# 678787 !)1% Π : 5x− y+ z− 5 = 0 ! Ω : x+ y+
2z − 7 = 08 1!:!$%& !)4%)/+#+ :#(%+!& '#+# x8 y ! z8 /#( 53!
%>!1!0#$ ?& !53#0=!&


5x− y + z = 5
x+ y + 2z = 7
⇒


y = 3x− 1
z = −2x+ 4
53! &*% #& !53#0=!& +!13<.1#& 1! 3$# +!/# r,
34!'5,# 678797 7#+# 1!/!+$.)#+ % '%)/% 1! .)/!+&!0*% 1#
101
 !"#$%
 !"# r$%& % '(#)% Ω* !& +,!
r :


x = −1 + 2t
y = 5 + 3t
z = 3− t
! Ω : 2x− y + 3z − 4 = 0,
%-.! /#&%. +,! +,#(+,! '%)"% 0! r 1 0#0% '% (x, y, z) =
(−1 + 2t, 5 + 3t, 3 − t)2 3 .! ,& '%)"% 0# !"# r "#&-1&
'! "!)$! #% '(#)% Ω* "!&%. +,!
2(−1+2t)−(5+3t)+3(3−t)−4 = 0⇒ −2t−2 = 0⇒ t = −1
3 .,-."4",4)0% )#. !+,#56!. '# #&1" 4$#. 0# !"# r7


x = −1 + 2(−1)
y = 5 + 3(−1)
z = 3− (−1)
⇒


x = −3
y = 2
z = 4
8% "#)"%* % '%)"% 0! 4)"! .!59% !)" ! # !"# r ! % '(#)% Ω 1
(−3, 2, 4)2 :!;# #4)0# +,! 2(−3)− (2)+ 3(4)− 4 = −6− 2+
12− 4 = 0 ⇒ (−3, 2, 4) ∈ Ω2
 !" #$%&'(
<!."# #,(#* 0!fi)4&%. # !+,#59% >! #( 0% '(#)% !* $%&% $%)?
.!+@A)$4#* "#&-1& 0!fi)4&%. %," #. B% &#. 0! !' !.!)"C?(%
#" #/1. 0! .,#. !+,#56!. /!"% 4#4. ! '# #&1" 4$#.2 D-% 0#?
&%. #(>,&#. 0#. .,#. ' %' 4!0#0!.* $%&% % E)>,(% 0! 0%4.
'(#)%. ! $%)0456!. 0! '# #(!(4.&% ! '! '!)04$,(# 4.&% !)?
" ! !"#. ! '(#)%.2 D(1& 04..%* #' !)0!&%. +,! # 4)"! .!59%
!)" ! 0%4. '(#)%. 1 !' !.!)"#0# '% ,&# !"# $%)"40# !&
#&-%.2
102
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
6
 !" 
 ! "#$%$&'&()
 !" #$%&'(! )*! &+)!,-. /!0! . /1!'. +)& 2.'%3* . /.'%.
P & 3 /&0/&'452)1!0 !. 6&%.0 +)& %&* 2.*. &7%0&*.8
4. /.'%.8 A & B '.8 8&9)5'%&8 2!8.8:
5; P = (0, 0, 0)< A = (1, 2, 3)< B = (2,−1, 2)=
55; P = (1, 1,−1)< A = (3, 5, 2)< B = (7, 1, 12)=
555; P = (3, 3, 3)< A = (2, 2, 2)< B = (4, 4, 4)=
56; P = (x0, y0, z0)< A = (x1, y1, z1)< B = (x2, y2, z2);
 $" >&%&0*5'& ! &+)!,-. 9&0!1 4.8 8&9)5'%&8 /1!'.8:
5; /!0!1&1. !. /1!'. Π : 2x − 3y − z + 5 = 0 & +)&
2.'%&'(! . /.'%. A = (4,−2, 1)=
55; /&0/&'452)1!0 ? 0&%!
r :


x = 2 + 2t
y = 1− 3t
z = 4t
& +)& 2.'%&'(! . /.'%. A = (−1, 2, 3);
 2" >&%&0*5'& . 6!1.0 4& m /!0! +)& 8&@! 4& 30o . A'9)1.
&'%0& .8 /1!'.8
Π1 : x+my+2z−7 = 0 & Π2 : 4x+5y+3z+2 = 0
 4" >&%&0*5'& . 6!1.0 4& n< 4& *.4. +)& .8 /1!'.8
Π1 : nx+y−3z−1 = 0 & Π2 : 2x−3ny+4z+1 = 0
8&@!* /&0/&'452)1!0&8;
 &" B&@!* A = (3, 1, 3)< B = (5, 5, 5)< C = (5, 1,−2) &
D = (8, 3,−6); C.8%0& +)& !8 0&%!8 AB & CD 8-.
103
 !"#$%
 !" !##$"%$& $ $" !"%#$ '() $*')+,! -)#) ! -.)"! *'$
)& !"%/(0
123 4 -.)"! Π !"%/( ! -!"%! A = (a, b, c) $ ) 56&%7" 6)
5) !#68$( ) Π /
√
a2 + b2 + c20 9" !"%#$ '() $*')+,!
5$&&$ -.)"!0
 !" #$%&'()*+$ ,-. -(+/+,-,&.
:!" .'6' ) )%6;65)5$ <= 9"%,! $"%$"5$' ) 5$fi"6+,! 5$ 9*')?
+,! 8$#). 5! -.)"!0 @$ #$&!.;$' ) *'$&%,! AB ;! C %#)D).E!'
) 5$fi"6+,! 5$ 9*')+,! -)#)(/%#6 ) 5! -.)"!0 9 )& )%6;65)?
5$& F $ GB !"&$8'6' $" !"%#)# '( #$&'.%)5! &)%6&2)%H#6! -)#)
$.)&= 9( )&! )fi#()%6;!B ;! C $"%$"5$' ! !" $6%! 5$ 7"8'.!
$"%#$ -.)"!&0 I)#) #$&!.;$# J $ KB / "$ $&&L#6! %$# '%6.6M)5! !&
 !" $6%!& 5$ #$%)& !"%65)& $( '( -.)"! $ 6"%$#&$+,! $"%#$
#$%)& $ -.)"!&0
 !0 1&2&*3'4+-.
@N9OPQRS:TB U.2#$5! B !"#!$%&' ()'*+$&,'0 @,! I)'.!B
V)W#!" Q!!W&B <XYZ0
[OVUB 9.!" [)8$& B !"#!$%&' ()'*+$&,' ! -*.!/%' 0&)!'%0
R6! 5$ \)"$6#!B OVIUB A]]J0
Q4[^ROPOB \!&/ ['6MB 1(*.!/%' 0&)!'% 0 @,! I)'.!B T)#D#)B
<XY]0
104