Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 62 стр.

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iº}ÈÏÈËã°mº
°
r
x
y
z
→
=
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p
→
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q
→
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rr
→→
−
0
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 ¹ºã ÈËä
rr p q
→→ → →
−= +
0
ϕθ
Ò°ãË
ºmÈËãÓº ¯ÈmÓËÓÒË ¹ãº°}º°Ò Ë
ÒäËmÒ
rr p q
→→ → →
=+ +
0
ϕθ

Ë
ϕθ
∈ −∞ +∞ ∈ −∞ +∞
(,); (,)

˺¯ËäÈº}ÈÏÈÓÈ

S

r
0
→

q
→

p
→

g
3
→

r
→
O  g
2
→

g
1
→
èqxytvr
˺¯ËäÈ

{°«}È«¹ãº°}º°mãº®Ë}ȯºmº®°Ò°ËäË}ºº¯ÒÓÈäºÎËÏÈ
ÈÓÈ¯ÈmÓËÓÒËämÒÈ
Ax By Cz D A B C
+++= + + >
00,

iº}ÈÏÈËã°mº
°ãºmÒË}ºä¹ãÈÓȯӺ°ÒmË}º¯ºm
rr
→→
−
0

p
→
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q
→
m}ºº¯ÒÓÈÓº®Áº¯äËÒäËË
m°Òã˺¯Ëä©mÒ
det
xx yy zz
ppp
qqq
xyz
xyz
−−−
=
000
0

|}È
Ax x By y Cz z
()()()
−+ −+−=
000
0
 ÒãÒ º}ºÓÈËãÓº
Ax By Cz D
+++=
0
 Ë Ò°ãÈ
A

B
Ò
C
ÓȲº«°« ¹º ˺¯ËäË  Ò ¯ÈmÓ©
°ººmË°mËÓÓº
A
pp
qq
B
pp
qq
C
pp
qq
yz
yz
xz
xz
xy
xy
==−=
det ; det ; det ,
È
DAxByCz
=− − −
000
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   iº}ÈÏȈËã°ˆmº                                                    S
                             x                                          
                          →                                                                                →                →
            ‚°ˆ   r = y  ÓË}ºˆº¯È« ˆº}È ÓÈ  r0  q 
                         z                                                        →

                                      →     →    → →
                                                            p
            ¹ãº°}º°ˆÒˆºÈmË}ˆº¯© p  q Ò r − r0        →                                 →
                                                           g 3  r 
          ­‚‚ˆ}ºä¹ãÈÓȯө cÒ° 
                                                          
          |ˆ}‚È m °Òã‚ ˆËº¯Ëä©  Ò ãËää© 
                               → →       →    →            
           ¹ºã‚ÈËä r − r0 = ϕ p + θ q  Ò °ãË                             →

          ºmȈËã Óº ‚¯ÈmÓËÓÒË ¹ãº°}º°ˆÒ ­‚ˈ
                                                           O    g 2 
                                                                     →
          Òäˈ mÒ
                         →    →     →     →
                                                            g1 
                         r = r0 + ϕ p + θ q              
                                                          
          Ë ϕ ∈ ( −∞ ,+∞ ) ; θ ∈ ( −∞ ,+∞ )          
                                                           èqxytvr
          
      ‘˺¯ËäȺ}ÈÏÈÓÈ



    ‘˺¯ËäÈ             {°«}È«¹ãº°}º°ˆ mã ­º®Ë}ȯˆºmº®°Ò°ˆËäË}ºº¯ÒÓȈäºÎˈÏÈ
                  ÈÓÈ‚¯ÈmÓËÓÒËämÒÈ
                                                   Ax + By + Cz + D = 0 ,                   A + B + C > 0 
            
            
     iº}ÈÏȈËã°ˆmº
      
                                                                      →    →       →        →
          °ãºmÒË}ºä¹ãÈÓȯӺ°ˆÒmË}ˆº¯ºm r − r0  p Ò q m}ºº¯ÒÓȈӺ®Áº¯äËÒäËˈ
          m°Òよ˺¯Ëä©mÒ
          
                                                             x − x0       y − y0       z − z0
                                                       det    px           py           pz      = 0 
                                                              qx           qy           qz
                                                                             
          
          |ˆ}‚È         A( x − x 0 ) + B ( y − y 0 ) + C ( z − z 0 ) = 0  ÒãÒ    º}ºÓȈËã Óº
           Ax + By + Cz + D = 0   Ë Ò°ãÈ A  B Ò C ÓȲº«ˆ°« ¹º ˆËº¯ËäË  Ò ¯ÈmÓ©
          °ººˆmˈ°ˆmËÓÓº
          
                                          py      pz                         px        pz                      px    py
                              A = det                    ;    B = − det                      ; C = det                       ,
                                          qy     qz                          qx        qz                      qx    qy
          
          È D = − Ax 0 − By 0 − Cz 0 ÒˆÈ}Ò亭¯ÈϺä䩹ºã‚ÒãÒˆº‚¯ÈmÓËÓÒ˹㺰}º°ˆÒ
          Ë°ˆ ‚¯ÈmÓËÓÒ˹˯mº®°ˆË¹ËÓÒ
          

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