Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 41 стр.

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cÈÏ Ëã
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c È Ï  Ë ã                                                      45
¯ºÒÏmËËÓÒ«mË}ˆº¯ºm



{©¯ÈÎËÓÒË°}È㫯Ӻº¹¯ºÒÏmËËÓÒ«m}ºº¯ÒÓȈȲ
                  
                  
                  
                                                                                                           →     →       →                                         →          →
                  ‚°ˆ ÏÈÈÓÈ°Ò°ˆËäÈ}ºº¯ÒÓȈ {O, g1 , g 2 , g 3 } ÒmÈmË}ˆº¯È a Ò b ã«}ºˆº
           →            →             →              →         →             →              →              →
¯©² a = ξ1 g1 + ξ2 g 2 + ξ3 g 3 Ò b = η1 g1 + η2 g 2 + η3 g 3 
        
        º°mº®°ˆmÈä°}È㫯Ӻº¹¯ºÒÏmËËÓÒ«
        
                      →→                         →             →              →            →               →                 →
                   (a , b )         = (ξ1 g1 + ξ2 g 2 + ξ3 g 3 , η1 g1 + η2 g 2 + η3 g 3 ) =
                                                   →      →                       →      →                       →       →
                                    = ξ1η1 ( g1 , g1 ) + ξ1η2 ( g1 , g 2 ) + ξ1η3 ( g1 , g 3 ) +
                                                     →     →                       →       →                         →       →
                                    + ξ2 η1 ( g 2 , g1 ) + ξ2 η2 ( g 2 , g 2 ) + ξ2 η3 ( g 2 , g 3 ) +                                                                                     
                                                     →     →                      →       →                          →   →
                                    + ξ3η1 ( g 3 , g1 ) + ξ3η2 ( g 3 , g 2 ) + ξ3η3 ( g 3 , g 3 ) =
                                          3                  →      →                       →       →                        →     →                3     3               →      →
                                    = ∑ ( ξ j η1 ( g j , g1 ) + ξ j η2 ( g j , g 2 ) + ξ j η3 ( g j , g 3 ) ) = ∑ ∑ ξ j ηi ( g j , g i ) .
                                          j =1                                                                                                    j =1 i =1
                                                                                                       
                                                                                           → → →
{°ã‚È˺¯ˆºÓº¯äÒ¯ºmÈÓÓºº­ÈÏÒ°È {e1 , e2 , e3 } ªˆÈÁº¯ä‚ãÈ‚¹¯ºÈˈ°«¹º°}ºã }‚ã«
¹º¹È¯Ó©²°}È㫯ө²¹¯ºÒÏmËËÓÒ®­ÈÏÒ°Ó©²mË}ˆº¯ºm°¹¯ÈmËãÒmº¯ÈmËÓ°ˆmº

                                                                                → →                  1, i = j
                                                                               (ei , e j ) = δ i j =          
                                                                                                     0, i ≠ j
                                               
Ë δLM ˆÈ} ÓÈÏ©mÈËä©® °Òämºã z¯ºÓË}˯È |ˆ}‚È ã« °}È㫯Ӻº ¹¯ºÒÏmËËÓÒ« mË}
ˆº¯ºmmº¯ˆºÓº¯äÒ¯ºmÈÓÓºä­ÈÏҰ˹ºã‚ÈËäÁº¯ä‚ã‚

                                                                              →→
                                                                           (a, b )= ξ 1η 1 + ξ 2η 2 + ξ 3η 3 
                                        
ÒÏ}ºˆº¯º®°ãË‚ ˆ¹ºãËÏÓ©Ë°ººˆÓº ËÓÒ«

                                      →
                                      a = ξ 12 + ξ 22 + ξ 32 

                                                  →        →          →        →                                     ξ 1η 1 + ξ 2η 2 + ξ 3η 3
                                    Èã« a ≠ o Ò b ≠ o  cos ϕ =                                                                                               .
                                                                                                               ξ12 + ξ 22 + ξ 32 η12 + η 22 + η32


       |ˆäˈÒ䈺¹º°ãËÓË˯ÈmËÓ°ˆmºm°ºˈÈÓÒÒ°‚°ãºmÒËä cos ϕ ≤ 1¹¯Òmº҈}
tnéjkntxzkyÇv¡qÈyt¹rvkxrvmv
       
                                          ξ1η1 + ξ 2η 2 + ξ 3η3 ≤ ξ12 + ξ 22 + ξ 32 η12 + η 22 + η32 ; ∀ ξ i ,ηi , i = 1,2,3. 
                                                                                                                 

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