03_Produto de Vetores_UFMG

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3&2-%& ', 12-'&2, :2'.2-&0$%.2)-25 E?'&4%&2.',# -%.?@.#
(.% 42,0:(%!4%42 -&0%):(!%&5
 !, -&'()%' ./0121&
,&fi./01" 23453 F%4', ', 12-'&2, ~u = x1~i + y1~j + z1~k 2
~v = x2~i + y2~j + z2~k# 42fi)0.', /(2 ' 6!"#$%" &(7*8*! C'(
6!"#$%" /.%&!." $($*8D# &23&2,2)-%4' 3'& ~u ·~v C-%.?@. @
0)40$%4' 3'& 〈~u,~v〉D# @ ' )6.2&' &2%!
〈~u,~v〉 = x1x2 + y1y2 + z1z2
+. 3%&-0$(!%&# ,2 ~u, ~v ∈ R2# 2. /(2 ~u = x1~i + y1~j 2 ~v =
x2~i + y2~j# ' 3&'4(4' 2,$%!%& fi$% 42fi)04' 42 H'&.% %)"!':%
I %)-2&0'&# 0,-' @#
〈~u,~v〉 = x1x2 + y1y2.
44
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
3
 !" 
34!'5,# 678727 !"#$ ~v = 3~i − ~j − 2~k ! ~w = ~i + ~j − ~k
%!&$'!( !) R
3
* +$#!)$( !(,'!%-./$( ,$)$* ~v = (3,−1,−2) !
~w = (1, 1,−1)* ! 0((1)
〈~v, ~w〉 = 〈(3,−1,−2), (1, 1,−1)〉 = 3 ·1+(−1) ·1+(−2) ·(−1)
⇒ 〈~v, ~w〉 = 4
9!fi+(;<# 672=7 2!"$)1"0)$( #!'>?@,# #! 3) %!&$' ~v =
(x, y, z)* '!+'!(!"&0#$ +$' |~v|* $ "4)!'$ '!0/ "5$ "!60&1%$*
|~v| =
√
〈~v,~v〉 789:;
<3! !) ,$$'#!"0#0( fi,0
|~v| =
√
x2 + y2 + z2.
>) R
2
* +$#!)$( #!fi"1' )?#3/$ #! )$#$ (1)1/0'* $3 (!@0*
#0#$ 3) %!&$' "$ +/0"$ ~u = (x, y)* (!3 )?#3/$ (!'A $ "4)!'$
'!0/ "5$ "!60&1%$
|~u| =
√
〈~u, ~u〉
* $3 01"#0 !) ,$$'#!"0#0(
, |~u| =
√
x2 + y2.
34!'5,# 678787
• !@0 ~v ∈ R3* ,$) ~v = (1, 0,−1) ⇒ |~v| =
√
12 + 02 + (−1)2 =
√
2.
• !@0 ~v ∈ R2* ,$) ~v = (−2,√5) ⇒ |~v| =
√
(−2)2 + (√5)2 =
√
9 = 3.
45
 !"#$%" #& '&%"!&( ) *!%& +
• !"#$" %! &' !($"! "#$% ~v ∈ R3& '%'( )(* ~v =
(1, 0,−1)& ( +#, -#*+(* ~w +#*. '%'( )(*
~w =
~v
|~v| =
1√
2
(1, 0,−1)
# %++/0&
~w =
∣∣∣∣ 1√2(1, 0,−1)
∣∣∣∣ =
∣∣∣∣( 1√2 , 0,− 1√2)
∣∣∣∣
=
√(
1√
2
)2
+ 02 +
(
− 1√
2
)2
=
√
2
2
= 1
1 -#*+(* '( -#*2(* ~v 3& 4% -#*'%'#& ,0 -#2(* ,4/2.*/(5
• )*#(+,-*. !,("! %$*# /$,($#! 6 '/+2748/% #42*#
'(/+ )(42(+ A = (x1, y1, z1) # B = (x2, y2, z2) 3 '%'%
)(*
d(A,B) =
∣∣∣−−→AB∣∣∣ = |B −A|
#& '#+2# 0('(&
d(A,B) =
√
(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2, A,B ∈ R3.
8(/48/'# 8(0 % '#fi4/:;( '# '/+2748/% #42*# '(/+ )(42(+
4( #+)%:(5 <%*% ( 8%+( '( )=%4(& >%+2%?4(+ +,)*/0/* %
2#*8#/*% 8((*'#4%'%& /+2( 3&
d(A,B) =
√
(x2 − x1)2 + (y2 − y1)2,
4#+2# 8%+(& A = (x1, y1), B = (x2, y2) )(42(+ '( R
2
5
 !"!# $%&'%()*+*), *& $%&*-.& /0.)%0&
<%*% @,%/+@,#* @,# +#$%0 (+ -#2(*#+ ~u = (x1, y1, z1), ~v =
(x2, y2, z2), ~w = (x3, y3, z3) #0 R
3
# λ ∈ R& 2%= @,#A
46
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
3
 !" 
 !" 〈~u, ~u〉 ≥ 0 # 〈~u, ~u〉 = 0 ⇔ ~u = ~0 = (0, 0, 0)$ ! "#$%&
'%() '%* +!fi-(./% 〈~u, ~u〉 ≥ 0& ! )! 〈~u, ~u〉 = 0& !-$/%
|~u| = 0 ⇔ ~u = ~00
 !!" 〈~u,~v〉 = 〈~v, ~u〉 !"#$%$&'%" 1!2# 34! 〈~u,~v〉 = x1x2+
y1y2 + z1z2 = x2x1 + y2y1 + z2z1 = 〈~v, ~u〉& '%() #) 5%%*6
+!-#+#) +! ~u ! ~v )/% -78!*%) *!#() ! 9#:!8 # 5%84$#6
$(9(+#+! +% '*%+4$% ! +# )%8# !8 R0
 !!!" 〈~u, (~v + ~w)〉 = 〈~u,~v〉+ 〈~u, ~w〉 (&)$*&+#$&'% ,!" *-.
/%01! 2 )!"% 3- '-$!*-)" ! "#$%& '%()
〈~u, (~v + ~w)〉 = 〈(x1, y1, z1), (x2, y2, z2) + (x3, y3, z3)〉
= 〈(x1, y1, z1), (x2 + x3, y2 + y3, z2 + z3)〉
= x1(x2 + x3) + y1(y2 + y3) + z1(z2 + z3)
= (x1x2 + y1y2 + z1z2) + (x1x3 + y1y3 + z1z3)
= 〈~u,~v〉+ 〈~u, ~w〉
 !%" 〈λ~u,~v〉 = λ〈~u,~v〉 = 〈~u, λ~v〉
34!$.-.(# 567626 & %#'!fi)*+,- .#/0* 1'-1'!#.*.# fi)*
)-2- *0!%!.*.# 1*'* %-)3$
 %" 〈~u, ~u〉 = |~u|2 ! "#$%& $!8%) 34! |~u| =
√
〈~u, ~u〉0 ;))(8&
(|~u|)2 =
(√
〈~u, ~u〉
)2
⇒ |~u|2 = 〈~u, ~u〉
34!'8,# 567656 |~u + ~v|2 = |~u|2 + 2〈~u,~v〉+ |~v|2 1*'* 45*!/6
45#' %#0-'#/ ~u~v ∈ R2 #/0* !75*8.*.# 0*29:2 : %;8!.* )*/-
-/ %#0-'#/ 1#'0#<+*2 *- R
3)$
47
 !"#$%" #& '&%"!&( ) *!%& +
 !"#$ %&!
|~u + ~v|2 = 〈~u + ~v, ~u + ~v〉
= 〈~u, ~u + ~v〉+ 〈~v, ~u + ~v〉 !"#$ !%&!%'"($(" '') " '''))
= 〈~u, ~u〉+ 〈~u,~v〉+ 〈~v, ~u〉+ 〈~v,~v〉 !&% '''))
= |~u|2 + 2〈~u,~v〉+ |~v|2
,&fi./01" 2345 !"#$% &' (%)* +',%-'*.3 !" ~u 6= ~0#
~v 6= ~0 " $" θ % & '()*+& ,&$ -".&/"$ ~u " ~v# "(.0&1
〈~u,~v〉 = |~u| |~v| cos θ 2345
6$.7 ,"fi(9:0& .7;<%; (0& ,"="(," ,7 >&(,9:0& ," &$ -".&?
/"$ "$.7/"; "; R
2
 (& =+7(&5 &* "; R
3
 (& "$=7:&53 @$$9;#
>7/& 7+*(&# % 9;=&/.7(." A*" -&>B 7."(." =7/7 & A*" % =/">9$&
C7D"/ >7$& A*"9/7;&$ &<."/ & '()*+& θ 7 =7/.9/ ,&$ -".&/"$ EF
>&(G">9,&$3
H"E7 A*" (7 "A*7:0& 2345.";&$ & $")*9(."
cos θ =
〈~u,~v〉
|~u| |~v| 2325
" 7$$9;# &<.";&$
θ = arc cos
( 〈~u,~v〉
|~u| |~v|
)
 23I5
67&89:" 23;3<3 J7/7 >7+>*+7/ & '()*+& "(./" &$ -".&/"$ ~u =
(1, 1, 4) " ~v = (−1, 2, 2)# C7:7;&$ & $")*9(." ;&-9;"(.&
cos θ =
〈~u,~v〉
|~u| |~v| =
〈(1, 1, 4), (−1, 2, 2)〉
|(1, 1, 4)| |(−1, 2, 2)|
cos θ =
−1 + 2 + 8√
18 · 3 =
9
9
√
2
=
√
2
2
⇒ θ = arc cos
(√
2
2
)
6 7$$9;# .";&$ A*" θ = 45o3
48
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
3
 !" 
 ! "#$%&'( %( )*+,$( #*-"# .(/0 1#-("#0 ~u # ~v2 3#"4#5#!(0
6,#7
8%9 〈~u,~v〉 > 02 4(! 5%0# *% #6,%&'( 8:;:92 -#!(0 6,# cos θ >
02 # %00/! 0 ≤ θ < 90o 8 (, 0#<%2 ,! )*+,$( %+,.(9;
859 〈~u,~v〉 < 02 3(" 8:;:92 -#!(0 6,# cos θ < 02 # %00/!
90o ≤ θ < 180o 8 (, 0#<%2 ,! )*+,$( (5-,0(9;
849 〈~u,~v〉 = 02 3(" 8:;:92 -#!(0 6,# cos θ = 02 # %00/!
θ = 90o 8 *#0-# 4%0(2 ,! )*+,$( "#-(9;
=/+,"% :;:>7 0 ≤
θ < 90o;
=/+,"% :;:?7 90o ≤
θ < 180o;
=/+,"% :;:@7 θ =
90o;
A50#"1# 6,# *% #6,%&'( 8:;:9 -#!(0 6,#
| cos θ| =
∣∣∣∣ 〈~u,~v〉|~u| |~v|
∣∣∣∣ ⇒ |〈~u,~v〉| = | cos θ| |~u| |~v| ,
# .#1#!(0 $#!5"%" 6,# 0 ≤ | cos θ| ≤ 1; B00/!2
|〈~u,~v〉| ≤ |~u| |~v| , 8:;C9
3%"% 6,%/06,#" 1#-("#0 ~u # ~v 80#<%! #$#0 3#"-#*4#*-#0 %( 3$%*(
(, %( #03%&(9;
34!'5,# 678797 A -"/)*+,$( D("!%.( 3#$(0 1E"-/4#0 A =
(2, 3, 1)2 B = (2, 1,−1) # C = (2, 2,−2) E "#-)*+,$(F
G%"% "#03(*.#"!(0 #0-% 6,#0-'(2 E /!3("-%*-# (50#"1%"!(0
0# %$+,! .(0 3%"#0 .# 1#-("#0 6,# .#-#"!/*%! (0 $%.(0 .(
49
 !"#$%" #& '&%"!&( ) *!%& +
 !"#$%&'( ABC )*( +,!+,$-".&'/!,)0 1/!/ "))(2
−−→
AB = (0,−2,−2)
−→
AC = (0,−1,−3)
−−→
BC = (0, 1,−1)
,$ *(2 ,3() 4&,
〈−−→AB,−→AC〉 = 〈(0,−2,−2), (0,−1,−3)〉 = 8
〈−−→AB,−−→BC〉 = 〈(0,−2,−2), (0, 1,−1)〉 = 0
1(! /$ (2 〈−−→AB,−−→BC〉 = 02 ") ( 52 $( 65! "., B2 () '/-() AB
, BC 7(!3/3 &3 #$%&'( -, 90o0 8))( $() +,!3" , .($.'&"!
4&, ( !"#$%&'( ABC 5 !, #$%&'(0
,-&./0" 123242 9, ,!3"$/! &3 6, (! (! (%($/' /() 6, (!,)
~u = (1,−1, 0) , ~v = (1, 0, 1)0 1/!/ "))(2 6/3() .($)"-,!/! &3
6, (! ~w = (x, y, z) (! (%($/' / ~u , / ~v0 :,$-( /))"32
〈~w, ~u〉 = 〈(x, y, z), (1,−1, 0)〉 = x− y = 0
〈~w,~v〉 = 〈(x, y, z), (1, 0, 1)〉 = x + z = 0
1(! /$ (2 $())( +!(;',3/ /%(!/ ), !,-&< / )('&."($/! ( )")=
 ,3/ 
x− y = 0x + z = 0 ⇒

x = yx = −z
8))( )"%$"fi./ -"<,! 4&, /) )('&?@,) )*( 6, (!,) $/ 7(!3/ ~w =
(x, x,−x)0 9,) , 3(-(2 ;/) /=$() ,).('A,! &3 !,+!,$),$=
 /$ , -,)),) 6, (!,)2 +(! ,B,3+'(2 (3/$-( x = 12 , /))"32
(1, 1,−1) 5 &3/ )('&?*(0
50
 !"#$!%! &!#'!"$() *+),-"(.)/ 0(1$# 2
3
 !" 
#$%$% &'()*+,( -* ./ 0*1('
3!fi+(56# 78298 !"#$ %& '!(%)!& ~u ! ~v* +%$ ~u 6= 0* ~v 6= 0
! θ % ,-./0% 1%)$#2% 3%) !0!&4 ! % '!(%) ~w )!3)!&!-(# #
3)%"!56% 2! ~u &%7)! ~v* 8 2!fi-:2% 3%)
proj ~v ~u =
(〈~u,~v〉
〈~v,~v〉
)
~v %/ proj ~v ~u =
(〈~u,~v〉
|~v|2
)
~v ;<4=>
?:./)# <4<@A 0 ≤ θ < 90o ?:./)# <4BCA 90o ≤ θ < 180o
D#-(% % ():,-./0% )!(,-./0% )!3)!&!-(#2% -# fi./)# ;<4<@>
E/#-(% % 2# fi./)# ;<4BC> -%& 3!)$:(!$ +%$3)!!-2!) E/!
|~w| = |~u| |cos θ| = |~u| |〈~u,~v〉||~u| |~v| =
|〈~u,~v〉|
|~v|
F%$% ~w ! ~v (G$ # $!&$# 2:)!56%* ~w = λ~v ! λ ∈ R* !-(6%*
|~w| = |λ| |~v| ⇒ |λ| = |~w| 1|~v| =
|〈~u,~v〉|
|~v|
1
|~v|
|λ| = |〈~u,~v〉||~v|2 ⇒ ~w =
( |〈~u,~v〉|
|~v|2
)
~v
H #&&:$* proj ~v ~u =
( |〈~u,~v〉|
|~v|2
)
~v4
51
 !"#$%" #& '&%"!&( ) *!%& +
,-&./0" 123242 !"#$ %&'&(")*!( proj ~v ~u &" +,& ~u =
(2, 3) & ~v = (1,−1)- ./$&(0& +,&
〈~u,~v〉 = 〈(2, 3), (1,−1)〉 = 2− 3 = −1
〈~v,~v〉 = 〈(1,−1), (1,−1)〉 = 1 + 1 = 2
1 !$$)"2
proj ~v ~u =
(〈~u,~v〉
|~v|2
)
(1,−1) = −1
2
(1,−1) = (−1
2
,
1
2
)
3#*$)%&(!*%# #$ 0&'#(&$ ~u & ~v 45&('&*6&*'&$ !# 57!*# #, !#
&$5!8#9 & ,$!*%# !$ 5(#5()&%!%&$ %& 5(#%,'# &$6!7!(2 '&"#$
+,&
〈~u + ~v, ~u + ~v〉 = 〈~u, ~u + ~v〉+ 〈~v, ~u + ~v〉
⇓
〈~u + ~v, ~u + ~v〉 = 〈~u, ~u〉+ 〈~u,~v〉+ 〈~v, ~u〉+ 〈~v,~v〉
6#"# |~u + ~v|2 = 〈~u + ~v, ~u + ~v〉2 |~u|2 = 〈~u, ~u〉 & |~v|2 = 〈~v,~v〉2
!7:" %& +,& 〈~u,~v〉 = 〈~v, ~u〉 45#)$ #$ !"/)&*'&$ ! +,& #$ 0&'#(&$
52
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
3
 !" 
 !"#!$%!& '() R
n
* %)& n = 2 )+ 3,- .''/&* 012!$3) 1'
3!4/31' '+5'#/#+/67!'* %8!91&)' 1
|~u + ~v|2 = |~u|2 + 2〈~u,~v〉+ |~v|2
⇓
|~u + ~v|2 = |~u|2 + 2| cos θ| |~u| |~v|+ |~v|2
≤ |~u|2 + 2|~u| |~v|+ |~v|2
≤ (|~u|+ |~v|)2
:)"#1$#)* )5#!&)'
|~u + ~v| ≤ |~u|+ |~v| ;<-=,
>'#1 3!'/9+1?313! @ %81&131 3! 3!%(45),6)6! 7$()+458
,)$- A!B1 )' !C!"%D%/)' ;E5 ! E3,-
 ! "#$%&'
F!'#1 1+?1* %)$8!%!&)' 1 3!fi$/6() 3! ")3+#) !'%1?1" ;)+
 ")3+#) /$#!"$), !$#"! 4!#)"!' ! '+1' ") "/!313!'- .?@&
3/'')* 4!"/fi%1&)' H+! ) +') 3) ")3+#) !'%1?1" !$#"! 3)/' 4!I
#)"!' !"&/#!I$)' !$%)$#"1" ) %)''!$) 3) J$9+?) !$#"! !?!'-
K!fi$/&)' 1 ")B!6() 3! +& 4!#)" ')5"! )+#") ! 1 3!'/9+1?I
313! #"/1$9+?1"-
 !( )*+,+-.-#$
;1, K13)' )' 4!#)"!' ~u = (2,−3,−1) ! ~v = (1,−1, 4)* %1?I
%+?1"L
/- 〈2~u,−~v〉M
53
 !"#$%" #& '&%"!&( ) *!%& +
 ! 〈~u + 3~v,~v − 2~u〉"
 ! 〈~u + ~v, ~u− ~v〉"
 #! 〈~u + ~v,~v − ~u〉!
$%& '() fi+,) -,), ./ #(0.)(/ ~u = (4,−1, 2) ( ~v = (−3, 2,−2)
,/ /(12 30(/ 4(/ 12,54,4(/6
 ! $ !"#$%&'(&(! (! )*+,&-.& |〈~u,~v〉| ≤ |~u| |~v|"
 ! $ !"#$%&'(&(! /-#&0$%'&-& |~u+~v| ≤ |~u|+ |~v|!
$+& 7).#( 82( 〈~u + ~v, ~u − ~v〉 = |~u|2 − |~v|2 -,), 82, /82()
#(0.)(/ ~u~v ∈ R2!
$4& 7).#( ,/ /(12 30(/ -).-) (4,4(/ 4. +.9-) 9(30. $.2
9:425.& 4( 29 #(0.)!
 ! |~v| = 0 /(; ( /.9(30( /(; ~v = 0"
 ! |~v + ~w| ≤ |~v|+ |~w|"
 ! |λ~v| = |λ| |~v|"
 #! |−~v| = |~v|!
$(& <,%(34. 82( |~u| = √2; |~v| = 3 ( 82( ~u ( ~v =.)9,9 29
>3125. 4(
3
4
π; 4(0()9 3(6
 ! |〈2~u− ~v, ~u− 2~v〉|"
 ! |~u− 2~v|!
$=& ?./0)( 82( , 4(fi3 @A. 4( >3125. (30)( #(0.)(/ -.4( /()
.%0 4, ,0),#(B 4, ,&- #"( ."((&/"( .%/()#,34. , fi12),
 !" #$% &$%%!'$%!
"#$ %&'()*#+, -#.,/ +01
2,/ $323$ a, b, c 40+3 0
'*#0+2023
a
2
= b
2
+c
2
−2bc ·cos θ,
/3)2, θ , ()*#+, 3)%&3
,/ /3*$3)%,/ 5#3 $31
23$ b 3 c6
$C!D&!
54
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
3
 !" 
 !"#$% &'()* θ + , -."#/, 0.1$0 ~u 0 ~v'
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