capitulo1_161
TRANSCRIPT
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Capítulo 1
!"#$%$&'( *+%"$&,%' -.'#/%' &0&12+&'
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Introdução3&45&' 676( 678( 679( 67: & 67;
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D2' +&'1+#"K#+&G2' L
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T&'/+#4Y2 $2 G2B#G&"12
> T&'/+&B&+ RE%Q#1%1#B% & RE%"1#1%1#B%G&"1&> !"#$%$&' $& G&$#$% NGE#12 #GO2+1%"1&
O+#G+%' '&45&' $2 U%O.1EQ2S
>
!"#$ !"&'( '\
> *+%"$&,%' O2$&G '&+ &'/%Q%+&'( B&12+&' ]
!^8' 2E !^8_ !^&'(
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Vetores3&45&' 976( 978 & 979
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0&12+&'
> T&'Q2/%G&"12( B&Q2/#$%$&
>
T&'Q2/%I'& 8 G&1+2'7 d%+% 2"$&\
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0&12+&'
> 0&12+&' 'Y2 K+%"$&,%' RE& 1&GGP$EQ2( $#+&4Y2 & '&"1#$2
> J O%+1./EQ% B%# $2 O2"12 J 2 O2"12 e
> FQ% '2[+&E EG $&'Q2/%G&"12
>
D&"_EG% #"[2+G%4Y2 '2@+& %1+%A&1P+#% N/2G2 &Q% [2#S
> W2$%' 1&G 2 G&'G2 $&'Q2/%G&"12
?+#&"1%4Y27
J
e
!
r
!
r
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f&O+&'&"1%4Y2 B&12+#%Q
> !G B&12+ C EG% '&1% 2+#&"1%$%
> f&O+&'&"1%4Y2 B&12+#%Q NJQKE"' Q#B+2' E'%G Q&1+% &G "&K+#12 !S
>
J1&"4Y2X
> + C EG &'/%Q%+ & &'/%Q%+ C GE#12 $#[&+&"1& $& B&12+g
>
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d+2O+#&$%$&' @i'#/%'
>
T # + & 4 Y 2 O 2 $ & ' & +&'O&/#k/%$% K+%k/%G&"1&N'&1% "2 &'O%42S
>
>
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d+2O+#&$%$& $% %$#4Y2
> F'/%Q%+&' '& '2G%G%QK&@+#/%G&"1&
> `:cKl:cK^abcK
>
32G% B&12+#%Q "Y2 C 1Y2'#GOQ&'X $#+&4Y2 & '&"1#$2#GO2+1%Gg
> !G% O%+1./EQ% '& $&'Q2/%$2 O2"12 J O%+% 2 O2"12e &( $&O2#' $2 O2"12 eO%+% 2 U
> hE%Q 2 $&'Q2/%G&"12+&'EQ1%"1& \
>
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J$#4Y2 $& B&12+&'
> f&K+% $2 1+#="KEQ2
> ? #"./#2 $2 '&KE"$2 "2
k"%Q $2 O+#G+2
> f&K+% $2 O%+%Q&Q2K+%G2
!
a
!
b
!
r =!
a +!
b
!
b
!
r
!
r
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d+2O+#&$%$&' $% %$#4Y2
> J '2G% C /2GE1%1#B% > J '2G% C %''2/#%1#B%
!
r
!
a
!
b
!
b !a
!
r =!
a +!
b =!
b +!
a!
a
!
b!
c
!
b +!
c
!
r =!
a +!
b+!
c( ) =!
a +!
b( )+!
c
!
r
!
a!
b
!
a +
!
b
!
c
!
r
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d+2O+#&$%$&' $&'E@1+%4Y2
> ? B&12+ C EG B&12+$& G&'G2' GP$EQ2 &$#+&4Y2 RE& G%'
'&"1#$2 2O2'12
> 3E@1+%4Y2 $& B&12+&'
> ?E
!
!
r
!
r
!
r
!
!
r
!
r + (!!
r ) = 0
!
c =
!
a !
!
b =!
a + !
!
b( )
!
a
!
b
!
!
b!
c
!
b
!
c
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0&12+
> !G% K+%"$&,% [.'#/% C EG B&12+ '& 2@&$&/& %&''%' O+2O+#&$%$&' %O+&'&"1%$%'
>
Fm&GOQ2' $& B&12+&'X O2'#4Y2( B&Q2/#$%$&([2+4%( G2G&"12 %"KEQ%+( 12+RE&( /%GO2&QC1+#/2( /%GO2 G%K"C1#/2( G2G&"12 $&$#O2Q2 &1/
>
J1&"4Y2X EG B&12+ C GE#12 $#[&+&"1& $& EG&'/%Q%+
!
r ! r
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Componentes e
vetor unitário
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F#m2' /%+1&'#%"2'
> U 2 G 2 1 + % @ % Q _ % +%QK&@+#/%G&"1& /2G 2'B&12+&'\
> ? GC12$2 K&2GC1+#/2 O2$&"Y2 '&+ O+&/#'2 2 'Ek/#&"1&
> !1#Q#,%+&G2' EG '#'1&G% $&/22+$&"%$%' & E'%+&G2' %'/2GO2"&"1&' $2' B&12+&'
"&''%' /22+$&"%$%'
> U 2 G O 2 " & " 1 & ' ' Y 2 % 'O+2A&45&' $2' B&12+&' "2m2
n
m
!
r
r x
r y
!
r x =
!
r cos!
r y =
!
r sen!
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U2GO2"&"1&'
> U2GO2"&"1&' O2$&G'&+ "&K%1#B%'
> < P $ E Q 2 & % '
/2GO2"&"&1&' 1&GE"#$%$&
n
m
!
r
r x
r y
!
r x =
!
r cos! < 0
r y =
!
r sen! !
r = r x
2+ r
y
2
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0&12+&' E"#1i+#2'
> d % + % ' & O % + % + %$&'/+#4Y2 $2 GP$EQ2$ % $ & ' / + # 4 Y 2 $ %$ # + & 4 Y 2 o ' & " 1 # $ 2
E1#Q#,%+&G2' 2' B&12+&'E"#1i+#2'
> !G B&12+ E"#1i+#2 C EGB&12+ $& GP$EQ2 6
> H"$#/% %O&"%' % $#+&4Y2& 2 '&"1#$2
> F G K & + % Q ( B % G 2 '$&'/+&B&+ 2' B&12+&'& G 1 & + G 2 ' $ 2 'E"#1i+#2' %2 Q2"K2 $2'
m2' E1#Q#,%$2'
û =
!
u
!
u
H"$#/% % E"#1%+#&$%$&$& EG B&12+
n
m
!
r
r x
r y
!
ˆ!
!
r x = r
x !̂
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J$#4Y2 &G 8T
> 3&A% 2 B&12+
>
U2GO2"&"1&' 'Y2#"$&O&"$&"1&'gggg
>
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Fm&+/./#2
678`S3Y2 $%$2' 2' B&12+&'
T&1&+G#"&
%S
? GP$EQ2 $&@S ? ="KEQ2 $& &G +&Q%4Y2 %2 m2 ?p
/S ? GP$EQ2 $2 B&12+
$S
? GP$EQ2 $2 B&12+
&S ? ="KEQ2 RE& 2 B&12+ [%, /2G 2 m2 ?q
!
a
!
a
!
a +
!
b
!
a !
!
b
!
a !
!
b
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Fm&+/./#2 678`
> %S
>
@S
>
/S
nNGS
m NGS
!
a
a y =!3
!
a x = 4
!
a = a x
2+ a
y
2
!
a = 42+ !3( )
2
= 16+9 = 5G
a x =
!
a cos!
cos! =4
5!! = arccos
4
5
"
#$
%
&' !! ( 37
o
!
r =
!
a +
!
b !
b
!
r = 102
+ !11( )2
= 100+121"
!
r !15m
!
r
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Fm&+/./#2 678`
> $S
> &S
nNGS
m NGS
!
a
tan! =s x
s y
!! = arctan 25
"#$ %
&' !! ( 22
o
!
s =!
a ! !
b
!
b
!
s = !2( )2
+5
2
= 4+25 "
!
s ! 5, 4m
!
s
!
s
! s y
s x
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FG 9 $#G&"'5&'
> ?' &m&GOQ2' $%$2'[2+%G &G 8 $#G&"'5&'
> FG 1+r' $#G&"'5&'
!
r = r x
2+ r
y
2+ r
z
2
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?O&+%45&' /2G B&12+&'
> 32G% $& B&12+&'
> 3E@1+%4Y2 $& B&12+&'
>
d+2$E12 &'/%Q%+> d+2$E12 B&12+#%Q N'P &G G2G&"12 %"KEQ%+S
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Produto Escalar3&4Y2 6b76 $2 Q#B+2 q2E"K %"$ -+&&$G%"?E 3&4Y2 ;76 N%O&"%' % $#K+&''Y2 '2@+& B&12+&'S
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d+2$E12 &'/%Q%+
> ? O+2$E12 &'/%Q%+&"1+& $2#' B&12+&' 1&G/2G2 +&'EQ1%$2 EG
&'/%Q%+> H " 1 & + O + & 1 % 4 Y 2
K & 2 G C 1 + # / % X C %O+2A&4Y2 $& EG B&12+
"% $#+&4Y2 $2 2E1+2B&,&' 2 GP$EQ2 $22E1+2
!
a
!
!
a ! !
b =!
a cos! !
b
!
a!
!
b
!
b
d+2A&4Y2 $2 B&12+ " "%$#+&4Y2 $2 B&12+ #
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d+2$E12 &'/%Q%+
> d+2A&4Y2 $& # "% $#+&4Y2$& "
> ? O+2$E12 &'/%Q%+ C
/2GE1%1#B2
> d+2$E12 &'/%Q%+ $& EGB&12+ O2+ &Q& G&'G2
> 3 & $ 2 # ' B & 1 2 + & '2+12K2"%#' N"^ab2S
!
a
!
!
b ! !
a =!
b cos! !
a
!
b
d+2A&4Y2 $2 B&12+ # "%
$#+&4Y2 $2 B&12+ "
!
a!
!
b =!
b!
!
a
!
a!
!
a =
!
a
!
a cos0=
!
a
2
=
:
!
a
2
!
a !
!
b =!
a
!
b cos!
!
a !!
b"!
a #!
b = 0
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d+2A&45&'
> d+2A&4Y2 "2' m2' /%+1&'#%"2'
> U2G2 2' m2' /%+1&'#%"2' 'Y2 2+12K2"%#'
!
a !
!
b =!
a
!
b cos!
!
a !
!
b"
!
a #
!
b =0
d $ 1 Q
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d+2$E12 &'/%Q%+ &G/2GO2"&"1&'
> d+2$E12 &'/%Q%+
> T&'&"B2QB&"$2
!
a ! !
b
b b
!
a ! !
b = a xb x + a yb y + a zb z
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Fm&+/./#2
3&A%G 2' B&12+&'
/2G %' /2GO2"&"1&' $%$%' &G E"#$%$&
%+@#1+i+#%%S hE%Q 2 ="KEQ2 &"1+& &''&' B&12+&'\
-
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Fm&+/./#2
> d+2$E12 &'/%Q%+ +&Q%/#2"% B&12+&' /2G 2="KEQ2 &"1+& &Q&'
>
W&G2' RE&> F RE&
>
d2+ 2E1+2 Q%$2
!
a !
!
b =!
a
!
b cos!
!
a = a x2
+ a y2
+ a z2
!
!
a = 1
2
+3
2
= 10!
b = b x2+b y
2+b z
2!
!
b = 22+ "3( )
2
= 13
!
a !
!
b = 10. 13 cos! = 130 cos!
!
a ! !
b = a xb x + a yb y + a zb z
!
a !
!
b =1.0+3.2"0.3= 6
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Fm&+/./#2
> d2+ EG Q%$2
> d2+ 2E1+2
>
d2+1%"12
> d2+ kG
!
a !
!
b = 130 cos!
!
a ! !
b = 6
130 cos! = 6! cos! =6
130=
18
65
! = arccos 18
65
!
"
#$
%
& '! ( 580