Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 70 стр.

UptoLike

Составители: 

Рубрика: 


Ë} ÒÒ}ÈÁË ¯©m©° Ë®äÈËäÈÒ}Òlnj
ÙkÓÈãÒÒ˰}È«˺äË¯Ò«ÒãÒÓË®ÓÈ«ÈãË¯Èµ
äÓºmkp
xx
xx
yy
yy
zz
zz
=
=
1
21
1
21
1
21

˰ãÒºã}º
()()()
xxyyzz
212121
0
−−

°

¯ÈmÓËÓÒË
¹¯«äº®m
®mË}º¯Óº®
Áº¯äË
¯«äÈ« m ¹¯º°¯ÈÓ°mË äºÎË ©ÏÈÈÓÈ }È} ãÒÓÒ« ¹Ë¯Ë°ËËÓÒ«
m²¹ãº°}º°Ë®
(,) (,)
nr d nr d
11 2 2
→→
==

Ë
Q
1
Ò
Q
2
ÓË}ºããÒÓËȯөËÓº¯äÈãÓ©ËmË}º¯©ªÒ²¹ãº°}º°Ë®
È
d
1
Ò
d
2
ÓË}ºº¯©ËÒ°ãÈ
jãÒ ÎË ˰ãÒ ÒÏm˰ÓÈ º}È
r
0
 ˯ËÏ }ºº¯ ¹¯º²ºÒ ÈÓÓÈ«
¹¯«äÈ« º ¯ÈÒ°mË}º¯ ãº® º}Ò ªº® ¹¯«äº® ºmãËmº¯«Ë
°ãËË®°Ò°ËäË¯ÈmÓËÓÒ®
(, )
(, )
nrr
nrr
10
20
0
0
→→
→→
−=
−=

jãÒm}ºº¯ÒÓÈÓº®Áº¯äË
Ax By Cz D
Ax By Cz D
1111
2222
0
0
+++=
+++=

°

¯ÈmÓËÓÒË
¹¯«äº®mº
®mË}º¯Óº®
Áº¯äË
¯«äÈ«m¹¯º°¯ÈÓ°mËäºÎË© ÏÈÈÓÈ¹¯Ò¹ºäºÒ°ãºmÒ«}ºã
ãÒÓËȯӺ°ÒmË}º¯ºm
a
Ò
rr
→→
0
º˰mmÒË¯ÈmÓËÓÒ«
[, ]
ar r o
→→
−=
0
ÒãÒÎË

[,]
ar b
→→
=
Ë
bar
→→
=
[, ]
0

{ º¯ºÓº¯äÒ¯ºmÈÓÓº® °Ò°ËäË }ºº¯ÒÓÈ
{, , , }
Oe e e
123
→→
ÈÓÓºË
¯ÈmÓËÓÒË¹¯«äº®m¹¯º°¯ÈÓ°mË¹¯ÒÓÒäÈËmÒ

det
eee
aaa
xyz
b
xyz
123
→→
=
ÒãÒ
az ay b
ax az b
ay ax b
yzx
zxy
xyz
−=
−=
−=

|äËÒä ºm¹º°ãËÓË® °Ò°ËäË °}È㫯ө² °ãºmÒ® ºã}º mÈ ¯ÈmÓËÓÒ« ÒÏ ¯Ë²
ÓËÏÈmÒ°Òä©Ë º˰ãºËÒÏ¯ÈmÓËÓÒ®«mã«Ë°«°ã˰mÒËäm²¯Ò²iË®°mÒ
ËãÓºäÓºÎÒm¹Ë¯mºË¯ÈmÓËÓÒËÓÈ
a
x
mº¯ºËÓÈ
a
y
Ò¯ËËÓÈ
a
z
Ò°ãºÎÒmÏÈËä
¹ºãËÓÓ©Ë¯ÈmËÓ°mÈ¹ºãËÓÓº¹¯Ò²ºÒä}ºÎ˰mmÒÈ
0=0
¹¯ÒÓ«m¹¯Òªºämº
 Ë }  Ò Ò    } È Á Ë  ¯ ©   m © °  Ë ®   ä È ˆ Ë ä È ˆ Ò } Ò   l n ‘ j 
ÙkÓÈã҈Ò˰}È«˺äˈ¯Ò«ÒãÒÓË®ÓÈ«ÈãË­¯ÈµäÓºmkp



                                                                    x − x1   y − y1   z − z1
                                                                           =        =         
                                                                   x 2 − x1 y 2 − y1 z 2 − z1
                           
                           ˰ãÒˆºã }º ( x 2 − x1 )( y 2 − y1 )( z 2 − z1 ) ≠ 0 
                           
                           
° ¯ÈmÓËÓÒË           ¯«äÈ« m ¹¯º°ˆ¯ÈÓ°ˆmË äºÎˈ ­©ˆ  ÏÈÈÓÈ }È} ãÒÓÒ« ¹Ë¯Ë°ËËÓÒ«
      ¹¯«äº®m
      ® mË}ˆº¯Óº®      m‚²¹ãº°}º°ˆË®
      Áº¯äË                                                     → →                         →      →
                                                               (n1 , r ) = d 1            (n2 , r ) = d 2 
                                  →        →
                           ËQ1 ÒQ2 ÓË}ºããÒÓËȯөËÓº¯äÈã Ó©ËmË}ˆº¯©ªˆÒ²¹ãº°}º°ˆË®
                           Èd1 Òd2ÓË}ºˆº¯©ËÒ°ãÈ
                           
                                                                                   →
                           jãÒ ÎË ˰ãÒ ÒÏm˰ˆÓÈ ˆº}È r0  ˯ËÏ }ºˆº¯‚  ¹¯º²º҈ ÈÓÓÈ«
                           ¹¯«äÈ« ˆº ¯È҂°mË}ˆº¯ ã ­º® ˆº}Ò ªˆº® ¹¯«äº® ‚ºmãˈmº¯«Ëˆ
                           °ãË‚ Ë®°Ò°ˆËäË‚¯ÈmÓËÓÒ®
                                                                            → → →
                                                                            (n1 , r − r0 ) = 0
                                                                            → → →              
                                                                           (n2 , r − r0 ) = 0
                           
                                                                               A1 x + B1 y + C1 z + D1 = 0
                           jãÒm}ºº¯ÒÓȈӺ®Áº¯äË                                                     
                                                                               A2 x + B2 y + C2 z + D2 = 0
                           
° ¯ÈmÓËÓÒË           ¯«äÈ«m¹¯º°ˆ¯ÈÓ°ˆmËäºÎˈ­©ˆ ÏÈÈÓȹ¯Ò¹ºäºÒ‚°ãºmÒ«}ºã
      ¹¯«äº®mº                                                →        →    →
      ® mË}ˆº¯Óº®      ãÒÓËȯӺ°ˆÒmË}ˆº¯ºm a Ò r − r0 ˆº˰ˆ mmÒË‚¯ÈmÓËÓÒ«
      Áº¯äË
                                                                               → →     →        →
                                                                              [ a , r − r0 ] = o 
                           ÒãÒÎË
                                                                     → →       →            →        → →
                            [ a , r ] = b Ë b = [ a , r0 ] 
                           
                                                                                                                   →   →    →
                           { º¯ˆºÓº¯äÒ¯ºmÈÓÓº® °Ò°ˆËäË }ºº¯ÒÓȈ {O, e1 , e2 , e3 }  ÈÓÓºË
                           ‚¯ÈmÓËÓÒ˹¯«äº®m¹¯º°ˆ¯ÈÓ°ˆm˹¯ÒÓÒäÈˈmÒ
                           
                                                       →       →      →
                                                 e1           e2      e3                  a y z − a z y = bx
                                                                            →             
                            det a x          ay      a z = b ÒãÒ  a z x − a x z = b y 
                                                  x            y       z                  a y − a x = b
                                                                                           x         y       z

        
        
|ˆäˈÒä ˆº m ¹º°ãËÓË® °Ò°ˆËäË °}È㫯ө² ‚°ãºmÒ® ˆºã }º mÈ ‚¯ÈmÓËÓÒ« ÒÏ ˆ¯Ë²
ÓËÏÈmÒ°Òä©Ëˆº˰ˆ ã ­ºËÒÏ‚¯ÈmÓËÓÒ®«mã«Ëˆ°«°ã˰ˆmÒËäm‚²¯‚Ò²iË®°ˆmÒ
ˆËã Óº‚äÓºÎÒm¹Ë¯mºË‚¯ÈmÓËÓÒËÓÈ a x mˆº¯ºËÓÈ a y Òˆ¯Ëˆ ËÓÈ a z Ò°ãºÎÒmÏȈËä
¹ºã‚ËÓө˯ÈmËÓ°ˆmȹºãËÓÓº¹¯Ò²ºÒä}ˆºÎ˰ˆm‚mÒÈ0=0¹¯ÒÓ«m¹¯Òªˆºämº