&$6$!,&
91/0
&
&
&
&
(
K /
u X -I UI 2I w X -% U% 2% . u w <
!
& &
u ⋅ w = -I-%ZUIU%Z2I2%
$039<
• F 0 0 . ) & &
& &
& &
u ⋅ w u $ w u , w
3!
&
&
& &
u w ) u × w 17 7 2
/
& & & &
•F
/! u ⋅ w = w ⋅ u /
•
.
1%
3
2
91
&
&
(
/
uX-IUI2I w X-%U%2%)θ=
)
& &
(
u w !
&
& &
u ⋅ w
& &
= u . w . cos θ& ≤ θ ≤ I'&\
w
θ
&
;
u
8/3
F=
/
0
=/
/
/
5
2
=θ/
)!& ≤ θ ≤ I'&\
& &
5 u ⋅ w
cos
& &
θ = &u ⋅ w&
u .w
& &
= u . w . cos θ/
!
8
=θ
/
)
&
&
$θ
=
/
u w )
/
θX
(u&], w& ) 9
=!
\
/
)
/0
!I'&\→
→π
,0
3C/KN&\%π
3
+
.=
!5*
:5
*:5
•
3+/=θ
/
)!
&
&
u
w
& &
& &
θX&\Ÿ u w )
/
&\XIŸ u ⋅ w i&
&
&
u bb w &
u
θ
&
w
&
&
& &
θXI'&\Ÿ u w )
/
I'&\XOIŸ u ⋅ w j&
&
&
u bb w %J
&
w
&
&
&
&
θXJ&\=
Ÿ u w )
u ⊥ w & &
J&\X&Ÿ u ⋅ w X&
•
&
u
&
w
& &
&jθjJ&\=Ÿθi&Ÿ u ⋅ w i&
θ
&
u
& &
J&\jθjI'&\=Ÿθj&Ÿ u ⋅ w j&
&
w
θ
&
u
$039<
•9
!
& &
u ⋅ w = 0 !
(C/
[
(F/
1θXJ&\c@
3!J&\X&d
& & & & & &
= 0 [
! i ⋅ j = j ⋅ k = i ⋅ k = 0 &
&
&
B
/
V-
! i XI&& j X&I& k X&&I
&
&
&
• /
0 0
/
/! 0 ⊥ ∀u & &
& &
/
! u ⋅ 0 = 0 0 ⋅ u
()
A
&
/
u
& & &
& &
& &
≠ 0 w ≠ 0 u ⋅ w = 0 ⇔ u ⊥ w 02
/
Q,$!
&
7(5/
u
& &
&
= 3i + 8k w = 4,
(
2, − 5
)
/
w& ⋅ u& & &
&
;(
1/
u !( u ⋅ u
K&
&
& &
= | u | 2 0( u ⋅ 0 = 0 &
&
&
&
& &
>($ | u | X% | w | XKI%&\=
/
u w u ⋅ w J(
/
=/0
%KI;%IOI(%%O%0
=;
&
K((=
/
v X%IOI
AB KIO%;L&OL
&
&
L(5
/
/
v1 XIOI& v2 XI&I
KI
Q&5 $!#
&
&
&
7(
/
)
!( v XIO%K w XLM%0( i &
&
j
&
& &
& &
!/1
)K
-
7 u ⋅ w /0)0
;(5/
u XII& w X&I&/
u ⋅ w 0
0
>($1
=/0
OIO%L;OLO%&(KO%I5
=
/0
;
J(F;()/0
=,7
I&(
; ( K(B
-=
=;(%IK;KKM(&LI
&
L((
1K&\=
/
u XI%
&
&
&
j &
&
&
M(5/
a X%I b XZ%OM% c X%'
/
/
a + b &
1
/
c
&
− a N(5
=
=/0
%IK;I&OI(OI%I
&
&
O($=
/
u X%IOI v XIOIZ%0πbK
7G(
/
MIM;LK%(OKO%I)/0
=
=
&
&
&
&
&
&
&
&
77(P/
/
= + M 1 − LA = + I + % 1 + LA 1
W
&
&
& &
7;(5
/
w /
u X%OIK w ⋅ u XOL%
&
7>(5
/
7
/
v X%OII
7J(F
=
=;(
QMI%IK(
AB ⋅ AC + BA ⋅ BC + CA ⋅ CB & &
&
&
&
&
& &
7K(5
/
/ Y / YXM / 0
-F2 / ⋅ G XN G = %1 + KA & &
&
& &
&
7L(5
/
v -F22 v ⋅ v1 XI& v ⋅ v2 XOM v1 X%KOI
&
v2 XIOI%
7M(5
=
!/1.K
91/A91:7NU
&
&
&
& & & & &
7N(5/
u XIO%OI v XOII w XOII
u ⋅ v = (u + v ) ⋅ w & &
& &
&
&
& &
7O((
8/
u Z v u O v Y u YXLY v YXK=
u v 0N&\
& &
k XLM\ k XJ&\J(M&K(QL( ± IM M(fONKg
7(C2
6;( ⋅ G XI>( ;
N( ˆ X
M ≅MI\= ;ˆ X
%L
≅ME\ (ˆ X
2 ≅E%\O(OL7G( ; ⋅ ;( = & 77(fOK%g
J
NK
42
§
©
7;(ONKOJ7>(C0 ¨ & I
%
·
¸ 7J(INJ7K(LK&OLK&7L(OIL&
%¹
I
7M(
-E&\7N(X%7O( KE IK ;.
-3T
+
//
2)./8
#.
:/
K%
Download

Produto Escalar - Milton Procópio de Borba