Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 43 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 

cÈÏ Ëã
47
¯ºÒÏmËËÓÒ«mË}º¯ºm

°iã«}ºããÒÓËȯӺ°ÒÓËÓãËm©²mË}º¯ºm
a
→
Ò
b
→
Ó˺²ºÒäºÒº°ÈºÓº
º©Ò²mË}º¯ÓºË¹¯ºÒÏmËËÓÒË©ãº¯ÈmÓºÓã
vmº®°mÈmË}º¯Óºº¹¯ºÒÏmËËÓÒ«
°
[,] [,]ab ba
→→ →→
=−
jtzqrvuuyzjzqktvxz°ãËËÒÏº¹¯ËËãËÓÒ«ÒÓË 
ËÓº°ÒÁÓ}ÒÒ
sin
ϕ

°
[,] [,]
λλ
ab ab
→→ →→
=
°ãËË ÒÏ º¹¯ËËãËÓÒ« mË}º¯Óºº ¹¯ºÒÏmËËÓÒ«Òºº
ÁÈ}È º mË}º¯©
[,]
λ
ab
→→
Ò
[,]ab
→→
º¯ººÓÈãÓ© ºÓº® Ò º® ÎË
¹ãº°}º°Ò¹¯ÒÓË}ºããÒÓËȯө²
a
→
Ò
b
→
Ò
λ
≠
0

°
[,][,][,]abc ac bc
→→→ →→ →→
+= +
lqxzéqiyzqktvxz
iã«º}ÈÏÈËã°mÈÒ°¯ÒÒmÓº°ÒmË}º¯Óºº¹¯ºÒÏmËËÓÒ«ÓÈäÓË
º²ºÒä©°ãËÒËm°¹ºäºÈËãÓ©Ëm˯ÎËÓÒ«
ËääÈ

°ÈÓ©mÈmË}º¯È
a
→
Ò
b
→
ÓÈÈãÈ}ºº¯©²ÓȲº«°«mºË®
º}ËÓÈº°Ò
l
ºÈ¯ËÏãÈ¹ºmº¯ºÈ°ää©mË}º¯ºm
a
→
Ò
b
→
ÓÈ
ºã
ϕ
mº}¯º°Ò
l
¯ÈmËÓ °ääË ¯ËÏãÈºm ¹ºmº¯ºÈ }Èκº ÒÏ
ªÒ²mË}º¯ºmmº}¯º°Ò
l
ÓÈºã
ϕ
m˯ÎËÓÒËãËää©ËäººÏÓÈÈ}È}
l

ϕ

ý ý ý
ΛΛΛ
ϕϕϕ
,,,
()
lll
ab a b
→→ → →
+= +

pº°¹¯ÈmËãÒmº°«°ÓÈÒÏ¯Ò°
èqxytvr
ËääÈ

p°ãÒ
e
→
=
1
 º mË}º¯
[, ]
pe
→→
¯ÈmËÓ ¯ËÏãÈ ¹ºmº¯ºÈ
¹¯ºË}ÒÒ
mË}º¯È
S
→
ÓÈ¹ãº°}º°¹Ë¯¹ËÓÒ}㫯ÓmË}º¯
e
→
mº}¯mË}
º¯È
e
→
ÓÈºã
π
2
¹ºÈ°ºmº®°¯Ëã}Ë
c È Ï  Ë ã                                                      47
¯ºÒÏmËËÓÒ«mË}ˆº¯ºm



                                                                                                                              →            →
       °iã«}ºããÒÓËȯӺ°ˆÒÓËӂãËm©²mË}ˆº¯ºm a Ò b Ó˺­²ºÒäºÒº°ˆÈˆºÓº
            ˆº­©Ò²mË}ˆº¯ÓºË¹¯ºÒÏmËËÓÒË­©ãº¯ÈmÓºӂã 
       
       
       
vmº®°ˆmÈmË}ˆº¯Óºº¹¯ºÒÏmËËÓÒ«
       
       
                            → →                   → →
                  ° [ a , b ] = − [ b , a ]  jtzqrvuuyzjzqktvxz°ãË‚ˈÒϺ¹¯ËËãËÓÒ«ÒÓË
                       ˈӺ°ˆÒÁ‚Ó}ÒÒ sin ϕ  
                  
                                 → →                  → →
                  ° [ λ a , b ] = λ [ a , b ]  °ãË‚ˈ ÒÏ º¹¯ËËãËÓÒ« mË}ˆº¯Óºº ¹¯ºÒÏmËËÓÒ« Ò ˆºº
                                                                                       → →                        → →
                           ÁÈ}ˆÈ ˆº mË}ˆº¯© [ λ a , b ]  Ò [ a , b ]  º¯ˆººÓÈã Ó© ºÓº® Ò ˆº® ÎË
                                                                                                       →          →
                           ¹ãº°}º°ˆÒ¹¯ÒÓË}ºããÒÓËȯө² a Ò b Òλ≠0 
                  
                  
                             →      → →               → →               → →
       ° [ a + b , c ] = [ a , c ] + [ b , c ]  lqxzéqiyzqktvxz 
       
       
       iã« º}ÈÏȈËã °ˆmÈ Ò°ˆ¯Ò­‚ˆÒmÓº°ˆÒ mË}ˆº¯Óºº ¹¯ºÒÏmËËÓÒ« ÓÈä ­‚‚ˆ ÓË
º­²ºÒä©°ãË‚ ÒËm°¹ºäºȈËã Ó©Ë‚ˆm˯ÎËÓÒ«
       
       
                                                                                                →            →
    ËääÈ                         ‚°ˆ ÈÓ©mÈmË}ˆº¯È a Ò b ÓÈÈãÈ}ºˆº¯©²ÓȲº«ˆ°«mº­Ë®
                                                                                                                                                                             →           →
                                   ˆº}ËÓȺ°Ò l‘ºÈ¯Ëς㠈Ȉ¹ºmº¯ºˆÈ°‚ää©mË}ˆº¯ºm a Ò b ÓÈ
                                   ‚ºã ϕ mº}¯‚ º°Ò l ¯ÈmËÓ °‚ääË ¯Ëς㠈Ȉºm ¹ºmº¯ºˆÈ }Èκº ÒÏ
                                   ªˆÒ²mË}ˆº¯ºmmº}¯‚º°Ò lÓÈ‚ºãϕ
          
     ˆm˯ÎËÓÒËãËää©­‚Ë亭ºÏÓÈȈ }È}l
         ϕ
                            Λ             →       →              Λ             →           Λ            →
         ýϕ ,l ( a + b ) = ýϕ ,l a + ý ϕ ,l b 
        
        
        pº°¹¯ÈmËãÒmº°ˆ «°ÓÈÒϯҰ
            
            èqxytvr
            
                  
                                                   →                                             → →
    ËääÈ                         p°ãÒ e = 1  ˆº mË}ˆº¯ [ p, e ]  ¯ÈmËÓ ¯Ëς㠈Ȉ‚ ¹ºmº¯ºˆÈ ¹¯ºË}ÒÒ
    
                                                          →                                                                                                             →
                                   mË}ˆº¯È S Óȹ㺰}º°ˆ ¹Ë¯¹ËÓÒ}‚㫯ӂ mË}ˆº¯‚ e mº}¯‚mË}
                                                →                         π
                                   ˆº¯È e ÓÈ‚ºã                           ¹ºÈ°ºmº®°ˆ¯Ëã}Ë
                                                                           2
                  

Страницы