Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 44 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
iº}ÈÏÈËã°mº

[, ]
pe
→→

π
2
2

α


Pr
e
p

e

S
èqxytvr
ËääÈº}ÈÏÈÓÈ
¯ºmËËä mË ¹ãº°}º°Ò ºÓÈ ÒÏ }ºº¯©²
¹¯º²ºÒ ˯ËÏ º}
O
 ºËË ÓÈÈãº
mË}º¯ºm
S
Ò
e
¹Ë¯¹ËÓÒ}㫯Ӻ
e
Èmº
¯È«¹¯º²ºÒ˯ËÏmË}º¯©
S
Ò
e

|¯ººÓÈãÓÈ«¹¯ºË}Ò«mË}º¯È
S
 ÓÈ¹ãº°
}º°¹Ë¯¹ËÓÒ}㫯Ó
e
ËãËÎÈÓÈ
ãÒÓÒÒ ¹Ë¯Ë°ËËÓÒ« ¹º°¯ºËÓÓ©² ¹ãº°}º°Ë®
Òº  ÈÒÏº¹¯ËËãËÓÒ«mË}º¯Óºº¹¯ºÒÏmË
ËÓÒ«°ãËË
[, ] sin cos( )pe p e p
→→
==
α
π
α
2

¹º°}ºã}
e
=
1
cÒ°
vã˺mÈËãÓºm¯È°°äÈ¯ÒmÈËäºä°ãÈË
[, ] Pr
,
pe p
e
e
→→
=
ý
π
2
ΛΛ

Ë
Pr
Λ
e
p
ºÏÓÈÈËº¯ººÓÈãÓºË¹¯ºË}Ò
¯ºmÈÓÒËmË}º¯È
S
ÓÈ ¹ãº°}º°¹Ë¯¹ËÓÒ
}㫯ÓmË}º¯
e

˹˯ º¹Ò¯È«°ÓÈm˯ÎËÓÒ«ãËääÒº}ÈÎËäÒ°¯ÒÒmÓº°
mË}º¯Óºº¹¯ºÒÏmËËÓÒ«
iº}ÈÏÈËã°mº°mº®°mÈ
°

p°ãÒ
co
→→
=
º
°
ºËmÒÓº°
co
→→
ºÈ

[,]||[,
||
]||Pr()
| |( (Pr Pr ) )
,
,
abc cab
c
c
cab
cab
c
c
c
cc
→→ →→
→→
+=+ = = +=
==+=
→→
(
2.4.2.)
(2.1.)
 
 !

ý
ý
ΛΛ
ΛΛΛ
π
π
2
2

== +=
==+=+
→→
→→ →→
(2.4.1.)
(2.4.2.)

 
ý ý
||( Pr Pr )
||([ ,
||
][,
||
]) [ , ] [ , ] .
,,
ca b
ca
c
c
b
c
c
ac bc
c
c
c
c
ΛΛ ΛΛ
ππ
22
vmº®°mºº}ÈÏÈÓº
 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



   iº}ÈÏȈËã°ˆmº
                                → →                     ¯ºmËËä mË ¹ãº°}º°ˆÒ ºÓÈ ÒÏ }ºˆº¯©²
      [ p , e ] 
                                                         ¹¯º²º҈ ˯ËÏ ˆº}‚ O  º­ËË ÓÈÈãº
                                                                        →           →                                       →
                         π                               mË}ˆº¯ºm S Ò e ¹Ë¯¹ËÓÒ}‚㫯Ӻ e Èmˆº
      2                                                                                    →        →
     2                        ¯È«¹¯º²º҈˯ËÏmË}ˆº¯© S Ò e 
                                                         
     α                                                                                         →
        →     →                      →                   |¯ˆººÓÈã ÓÈ«¹¯ºË}Ò«mË}ˆº¯È S Óȹ㺰
      Pr → p  e                                                                      →
           ⊥e
                                                         }º°ˆ ¹Ë¯¹ËÓÒ}‚㫯ӂ  e ­‚ˈãËÎȈ ÓÈ
                             →
                                                         ãÒÓÒÒ ¹Ë¯Ë°ËËÓÒ« ¹º°ˆ¯ºËÓÓ©² ¹ãº°}º°ˆË®
      S                          ÒˆºÈÒϺ¹¯ËËãËÓÒ«mË}ˆº¯Óºº¹¯ºÒÏmË
     
                                                         ËÓÒ«°ãË‚ˈ
                                                        

                                                                → →              →     →                    →               π
     èqxytvr                         [ p , e ]    =      p    e    sin    α    =    p      cos   (     − α ) 
                                                                                                                             2
                                                                              →
                                                        ¹º°}ºã }‚ e = 1  cÒ° 
                                                        

                                                        vã˺mȈËã Óºm¯È°°äȈ¯ÒmÈËäºä°ã‚ÈË
                                                                                 → →               Λ        Λ         →
                                                                               [ p , e ] =  ý      π  →  Pr     → p 
                                                                                                        ,e       ⊥e
                                                                                                     2
                                                                 Λ      →
                                                        Ë Pr → p ºÏÓÈÈˈº¯ˆººÓÈã Ӻ˹¯ºË}ˆÒ
                                                                   ⊥e
     
                                                                                          →
                                                        ¯ºmÈÓÒË mË}ˆº¯È S  ÓÈ ¹ãº°}º°ˆ  ¹Ë¯¹ËÓÒ
                                                                                           →
                                                        }‚㫯ӂ mË}ˆº¯‚ e 
     ËääȺ}ÈÏÈÓÈ
           
           
           ‘˹˯ º¹Ò¯È«° ÓÈ‚ˆm˯ÎËÓÒ«ãËääÒº}ÈÎËäÒ°ˆ¯Ò­‚ˆÒmÓº°ˆ 
mË}ˆº¯Óºº¹¯ºÒÏmËËÓÒ«
           
   iº}ÈÏȈËã°ˆmº°mº®°ˆmȰ
     
                    →      →                                            →    →
            p°ãÒ c = o ˆº°ºËmÒÓº‚°ˆ  c ≠ o ˆºÈ
                                                            →
                            →    → →        →   →    →       c                                        →            Λ                    Λ          →        →
                           [ a + b , c ] = |c | [ a + b ,    →     ] = (   2.4.2.) = |c | ý π → Pr                                →    ( a+ b ) =
                                                                                                                               ,c        ⊥c
                                            | c|                                                           2
                                                                                                                                                                     
                                                                                        →             Λ                        Λ        →          Λ       →
                                         = (   !  2.1.) = |c |( ý π → (Pr                                    →   a + Pr         →    b) ) =
                                                                                                                   ,c          ⊥c                  ⊥c
                                                                                                               2
                                                                            →      Λ                 Λ        →               Λ                    Λ       →
                                         = (   2.4.1.) = |c |(ý π → Pr                       →   a + ý π → Pr                         →   b) =
                                                                                                 ,c       ⊥c                                 ,c        ⊥c
                                                                                             2                                           2
                                                                                        →                      →
                                                            →   →                   →                             → →                → →
                                                                                                                                                                         
                                                                                         c                     c
                                         = (   2.4.2.) = |c |([ a ,             →        ] +[ b ,      →        ]) = [ a , c ] + [ b , c ]                   .
                                                                                        |c |                |c |
        vmº®°ˆmºº}ÈÏÈÓº