Transcript Skalärprodukt, ON

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ced%fhgag7djiZd%fhkg;ljm;fhm
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0≤θ≤
p~
ln;o
iZqrisg;fhm=d c
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l no
p
u
vp
θ
u
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π €~
u·v p
u·v=
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fhm›tsg7g
l no
u
p™sš
fhm7d%fhmt›tsg7galjz
u 6= 0, v 6= 0
fhqeq fhm
u=0
v=0
ljz
m{jh“œ;%u•9‰jŠe‰už*ln;o
v ™
i
qeq fhmtsy+g7cey
u ⊥ v p š+™
zfhqeqrtsy
l no
m g
θ
u
v ™ ~
p
mad ec yxk%fhqey
™
ljz
|u||v| cos θ
0
km7ced%fhm
žGljzŸvxfhg
qeq fhm›tsg7g
u ⊥ v
u · v = 0.
p
š+™
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m y‚ljqeqed%fhkg;ljm7yžbfhqeq h
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š
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u’
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p
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k y‚fhqrt
tsm¸
š
(αu)·v = α(u · v).
™
z
•
q ·;tsy‚vxfm
³
u·v=v·u
~
ln;o
qeq fhm=¶
~
fhkg;ljm;fhm7ybt
•
š+™
u · u = |u|2
~
º
¼
kutsq
v
m=´btsm7d c
e1 , e 2 , e 3
p
ljm7g;l
š
cBm7µxzwzfhg
ljybtsqrt
u = (x1 , x2 , x3 )
ln;o
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µxg
mt½fhy¾jh“œj•9j€«›”€¥‰+‘®¿;‰s†AljzÀvxfwobtsmq y vxfhy
p™sš%p
š%³
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g=cÁ´xqrtsy‚fhg By
ijv‚tsy{bt
kutsqeqrtsm=d cE¶ mwÂÃÄ7¿7‰s†
š
~
p
p
³
~
v = (y1 , y2 , y3 )
z
~
t
~
´
~
fhyƒ
=Ŝbt
i
pp
™
¹
m
u · v = x 1 y1 + x 2 y2 + x 3 y3
q
|u| = x21 + x22 + x23 .
fhm7d%fhmtÆtsg7gcafhyV aŜbt
m›k%l ljm;v}ceybtsg;fhm7ybt¯¶ mfhy¾d%fhk g;ljmAo‚fhqeg›fhyxk%fhqeg›q y vxfhyÇtdÈvxf
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qeqeg i}ÉGljz
cDfhy
u = x 1 e1 + x 2 e2 + x 3 e 3
™
p
~
p
aŜbt
i
qeq fhmtsg7g
l no
p e1 , e 2 e3 p
š+™
x1 = u · e 1 , x 2 = u · e 2
x3 = u · e 3 .
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f
š
p
y‚v¢lj fhm7djtsg7c ljy
™
p
td¢d%fhk g;ljm7y
™
m=tsg7gDfhy¢y‚ljm7z›tsqed%fhk g;ljmag7ceqeqEfhg7gR´xqrtsy
π : ax + by + cz + d = 0
n = (a, b, c).
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ced%fhgZ´xqrtsy‚fhg
k%ljzw´El
p
tsy g;fhm
žÓvxfhqrt{µx´x´Æd%fhk g;ljm7y
π : ax + by + cz + d = 0
ln;o
ijv‚tsybttsg7g
u1
u2 p
u1 k π
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´xm;ls·7fhkg7c ljy‚fhy¡td
u
´bi
n.
l no
u = (x0 , y0 , z0 )
cdceyxk%fhqem
™
gt
u2 ⊥ π.
f
t dd%fhkg;ljm7y

š p
n=
m;ls·7fhkg7c ljy §¶ ljm7zfhqey
fhmatsg7g
å
p
š
(a, b, c).
ãisg
u2
djtsmtHvxfhy¢ljm7g;l
š
ljybtsqrt
kjtsq
=
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f
ç
g
p
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=Ŝbt
fhm
p
u·n
n
|n|2
x0 a + y 0 b + z 0 c
(a, b, c).
a2 + b 2 + c 2
u2 =
•
m7´xm;lv}µxk gžM
™
³
q ·7fhm
º
~
td¢¼
u
2
u
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š
cv
n
u
1
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p
p
p
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u1 = u − u 2
š
fhqexceq vxfhy¾tdê´xµxyxkg;fhy
P : (x0 , y0 , z0 )
cR´xqrtsy‚fhg
π : ax + by + cz + d = 0.
ÝÞbß-à\á§àãâãä*yy‚ljm7z›tsq‚g7ceqeqx´xqrtsy‚fhg
A = (x1 , y1 , z1 ) ∈ π.
ç
π
ceq v‚t—d%fhkg;ljm7y
š
f
td—d%fhkg;ljm7y
t
ífhqrt{µx´x´îvxfhyxybt{d%fhk g;ljmZc*dceyxk%fhqem
k%ljzw´El
gt
™
ljzïc9f€ðxfhz—Å
ln;o
tsy g;fhm
|
lv}g0ì}n;kqec
š 
´xµxyxkg
P
u2 p
u = u 1 + u2
v m
l no
zòdc
™
u1 k π
u2 ⊥ π.
l no
kutsqeqrtsmD´xm;ls·7fhk g7c ljy ´xµxyxkg;fhy^¶ m
Q
p
³
´Mf fhqe´xµxyxkg;fhyó¶ m
i½¶¨i
y µÈzfv
p
š
³
p
S p
d%fhkg;ljmtjvxv}ceg7c ljyétsg7g
~ = OP
~ − u2
OQ
ln;o
~
~
OS = OP − 2u2 .
u1
p
´xq fhg¢ldutsyÉñd =
c obtsm¢v‚i
fhy
n = (a, b, c). ë š
š
~
u = AP = (x0 − x1 , y0 − y1 , z0 − z1 ).
p
u2
u
O
A
u1
n
Q
S
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f
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ç
g
p
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™
p
š
ced%fhq h
f m
p
™
maz
~
t
~
´
~
fhy
gisy‚vxfhgDzfhqeqrtsy½´xµxyxkg;fhm7ybt
š
ced%fhy¡
aŜbt
p
P : (x1 , x2 , x3 )
e1 , e 2 , e 3 .
l no
Q : (y1 , y2 , y3 ).
ÝÞbß-à\á§àãâãä
ãd
Óisg
vxfhg
u = P~Q = (y1 −x1 , y2 −x2 , y3 −x3 ).
mãv‚iRqecekutRzfvq y vxfhyZtdd%fhk g;ljm7y
žv}d
u
™
™ š
zfv
|u| =
•
ç
f
p
g
™
p-~
gisy}Å
u
qecekjt
ÝÞbß-à\á§àãâãä
gisy‚vxfhg=zfhqeqrtsy½´xµxyxkg;fhy
p
ë
t
š
fhy
š
l v}gìnk qec
š
ln;o{qeceyj·7fhy
P : (x1 , y1 , z1 )

 x = x0 + tα
y = y0 + tβ
l:

z = z0 + tγ
´xµxyxkgA´biêqeceyj·¥fhy
fhm
g f€ð
´xµxyxkg;fhy
ec q v‚t
~
~
P0 : (x0 , y0 , z0 ). ç
k ybt
f v‚tsy—vxfhyZljm7g;l ljybtsqrt ´xm;ls·7fhkg7c lj‚

y fhy
™
p
š
l,
u = P~0 P = (x1 −x0 , y1 −y0 , z1 −z0 ). ç
´bi—qeceyj·¥fhy
m7cek g7yxcey
d%fhkg;ljm
p
š%p
u
v = (α, β, γ).
tdýd%fhk g;ljm7y
u0
Q
P
p
(y1 − x1 )2 + (y2 − x2 )2 + (y3 − x3 )2 .
zütd
d%fhkg;ljm7y
p
kjtsq
m;ls·¥fhkg7c ljy
å
¶¨ljm7zfhqey
p
u0 =
fv‚tsyþEf
oj·
qe´ÿtd
™
Ód
f
•
gisy‚vxfhg
p
ç
g
p
g
p
u·v
v.
|v|2
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P0 = (x0 , y0 , z0 ) ∈ π.
f
•
gisy‚vxfhg
0
P : (x1 , y1 , z1 )
π
ceq v‚t—d%fhk g;ljm7y
ç
f
š
ln;o{´xqrtsy‚fhg
π : ax + by + cz + d = 0.
td¢d%fhk g;ljm7y
t
ç
g
p
zütd
™
lv}g0ì}n;kqec
ceq v‚twvxfhga´xqrtsy
ç
ljz
p
mZ´btsmtsqeq fhqeqeg—zfv
n
u’
P0
|u0 |.

 x = x2 + tα2
y = y2 + tβ2
l2 :

z = z2 + tγ2
ln;o
π : ax + by +
l no
ljz
l1
p
P
ceyxy‚fhobisqeq fhm
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gisy‚vxfhgHzfhqeqrtsy
p
my+µ®qecekut›zfv‡td gisy‚vxfhgzfhqŒÅ
™
p
qrtsyZ´xqrtsy‚fhg
l noZfhy
l v}gìnk qec ´xµxyxkgã´bi
š
š
π
fhm k ybtÆvxfhg7gtótd gisy‚v zfv¾zfhg;lsÅ
l2 . ç
™
p
vxfhy
ljz
Mf k m7cedceg
ldutsy
p
p
p
~
l2 .
™
eq ceyj·7fhm7ybt
u
l2
u’
P0
π
l1
êÑÒ*ÏÎ9ۜøDÎb×2ÐuùÁúVÒñÑ0ÒÚÜEË
f
ç
p
g
™
z–dceyxk%fhqey{zfhqeqrtsy{d%fhkg;ljm;fhm7ybt
ÝÞbß-à\á§àãâãäÁByxqec
š
g
p
kutsq
™
m7´xm;l v}µxkg;fhy
u = (x1 , y1 , z1 )
p
vxf€„‚yxceg7c ljy
™
cos θ =
æ¡fhy½f€¶§g;fhm
p
´xµxyxkg
P

 x = x1 + tα1
y = y1 + tβ1
l1 :

z = z1 + tγ1
cz + d = 0
•
š
gisy‚vxfhg=zfhqeqrtsy½qeceyj·7fhm7ybt
p
ÝÞbß-à\á§àãâãä
fhy
n = (a, b, c). ë š
š
~
u = P0 P1 = (x1 − x0 , y1 − y0 , z1 − z0 ).
p
u
mDy+µ½qecekjtZzfv
™
td¢¼
u’
P
u·n
u0 =
n.
|n|2
p
»
~
u’’
fhm k ybt
f v‚tsyîvxfhyÈljm7g;l ljybtsqrtƒ´xm;lsÅ

ç
™
p
š
·7fhkg7c lj‚
y fhy
tdHd%fhkg;ljm7y
´bia´xqrtsy‚fhg
p
u0
u
m;ls·¥fhkg7c ljy ¶¨ljm7zfhqey
fhm
y‚ljm7z›tsq
n. å
p
š
Ód
cv
p
u
zfv
gisy‚vxfhg=zfhqeqrtsy½´xµxyxkg;fhy
p
fhm
p
P
u00
u = u0 + u00 ~
mDy+µ½qecekjtZzfv
|u00 |.
™
zütd
™
=Ŝbt
fhm
š
z
d%fhk g;ljm7y
™
p
tszbtsy‚vxfhg
p
m7´xm;lv}µxk gžM
™
ljz
dcãtsm7Mfhgtsm=cfhy®
=Ŝbt
u · v = x 1 x2 + y 1 y2 + z 1 z2
q
|u| =
x21 + y12 + z12
q
x22 + y22 + z22
|v| =
p
kutsy{o
³jš
m
l no
v = (x2 , y2 , z2 ).
zfhy½v}d
u · v = |u||v| cos θ,
u·v
|u||v|
fhm7q fvxfhgMfhm
u
θ
v
™
kybt
p
fhyxqec
š
g
p~
kjtsq
qeqeg
•
ç
fv‚tsy{kutsy^d ceyxk%fhqey
p
f
g
p
m7´xm;lv}µxk gžM
=Ŝbt
p
fhm
p
θ
fhyxk%fhqeg=Efhm
™
kybt
p~
™

 x = x1 + tα1
y = y1 + tβ1
l1 :

z = z1 + tγ1
ífhy
ln;o

 x = x2 + tα2
y = y2 + tβ2
l2 :

z = z2 + tγ2
l
1
v1
θ
l2
v2
f
g
p
z–dceyxk%fhqey{zfhqeqrtsy
™
ÝÞbß-à\á§àãâãäÁí
fhy
p³
n1 = (a1 , b1 , c1 )
ceg
•
ldutsy
p
f
g
π1 : a1 x + b 1 y + c 1 z + d 1 = 0
k gtZdceyxk%fhqey
l no
™
ln;o
p
n2 = (a2 , b2 , c2 ).
y‚ljm7z›tsqed%fhkg;ljm;fhm
p
ífhyxybtdceyxk%fhqxEfhm k ybt
zfvwzfhg;l vxfhy
ljz`Ef k m7ced Å
™
p
p
p
~
z–dceyxk%fhqey{zfhqeqrtsy{´xqrtsy‚fhg
™
ÝÞbß-à\á§àãâãä
dceyxk%fhqey
yxcey
θ
f g z
ç
p ™
zfhqeqrtsy qeec yj·¥fhy
d%fhk g;ljm
p
¶
m g
³ p
m7cek ¥
g Å
l no
v = (α, β, γ)
y‚ljm7z›tsq
p
n = (a, b, c).
ífhy
k gtdceyxk%fhqey
f
fv‚tsy
α š pp
p³
td
α = | π2 − θ|.
š p
%
´xqrtsy‚fhg
π2 : a2 x + b2 y + c2 z + d2 = 0.
m=qecekutZzfv^dceyxk%fhqey{zfhqeqrtsy{´xqrtsy‚fhy
π : ax + by + cz + d = 0
l : (x, y, z) = (x0 , y0 + z0 ) + t(α, β, γ).
ç
¼›td¢¼
x1 x2 + y 1 y2 + z 1 z2
p
cos θ = p 2
,
x1 + y12 + z12 x22 + y22 + z22
kgt?dceyxk%fhqey
m
ï
qec΁
p;³
™
kut zfvïdceyxk%fhqey
zfhqeqrtsy
qeceyj·7fhm7ybt
m cek Å
7
p
g7yxcey
d%fhk g;ljm;fhm
ln;o
š%p
v1 = (α1 , β1 , γ1 )
ífhyxybt^dceyxk%fhqEfhm k+Å
v 2 = (α2 , β2 , γ2 ).
™
ybt
zfv^zfhg;l vxfhy
ljz–Ef k m7cedceg
ldutsy
p
p
p
p
~
ç
~
z–dceyxk%fhqey{zfhqeqrtsy{qeceyj·7fhm7ybt
ÝÞbß-à\á§àãâãä
•
cv
iZxqecem
p
ln;o
™
θ
α
n
ln;o½qeceyj·¥fhy