Аналитическая геометрия. Шурыгин В.В. - 48 стр.

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1
◦
2
◦
{O; e
i
} E
2
ha, bi = ha
i
e
i
, b
j
e
j
i = a
i
b
j
he
i
, e
j
i = ε
ij
a
i
b
j
= ε
12
a
1
b
2
+ ε
21
a
2
b
1
ε
ij
=
he
i
, e
j
i
ha, bi = ε
12
a
1
a
2
b
1
b
2
.
ha, bi = |a||b|sin ϕ,
ϕ = ϕ
b
− ϕ
a
a b
ha, bi = ([a], b) = |a||b|cos(ϕ
a
+
π
2
− ϕ
b
) = |a||b|sin(ϕ
b
− ϕ
a
).
|ha, bi| a b
ha, bi > 0 ⇐⇒ {a, b}
ha, bi < 0 ⇐⇒ {a, b}
ha, bi = 0 ⇐⇒ a||b
ABC
S
4ABC
=
1
2
|h
−→
AB,
−→
AC i| =
1
2
mod
x
B
− x
A
y
B
− y
A
x
C
− x
A
y
C
− y
A
.
ABC h
−→
AB,
−→
AC i > 0
A 7→ B 7→ C 7→ A
h
−→
AB,
−→
AC i < 0
A 7→ B 7→ C 7→ A
   H-7?[29/? /b-M/8b6;0 1 0 2 \ 7*:1- .-72)08? O-3;272 L79 b)0/7*K
                                           ◦       ◦
,09 1-/-:- .3-0[b*L*,09 b .3-0[b-7?,-; 6OO0,,-; 3*.*3* {O; e } ,6 E †
ha, bi = ha e , b e i = a b he , e i = ε a b = ε a b + ε a b \ :L* ε =
                                                                                                     i        2
                 i       j           i j                         i j            1 2        2 1

he , e i = „610; -+36[-;\
                     i       j                 i   j        ij             12         21                 ij

 i    j


                                                             a1 a2
                                       ha, bi = ε12                    .
                                                             b1 b2
     xuµxzTWñx¶yWx ¶tu̶ztÍ yu¶uVu ³TuWêtxwxXW´Z
   ‡79 1-/-:- .3-0[b*L*,09 0;**8/9 /7*L2J}69 O-3;276\ 6,67-:0),69
O-3;27*\ .- 1-8-3-M -.3*L*79*8/9 /16793,-* .3-0[b*L*,0* b*18-3-b†
                         ha, bi = |a||b| sin ϕ,                   ‚Æ•ƒ
:L* ϕ = ϕ − ϕ · 2:-7 Šž ”Ž‰žŠ a ªŠ ”Ž‰žŠ b =
   ‡*M/8b08*7?,-\
             b           a



                                                             π
          ha, bi = ([a], b) = |a||b| cos(ϕa +                  − ϕb ) = |a||b| sin(ϕb − ϕa ).
   —[ O-3;27 ‚Æ•ƒ b8*16J8 /7*L2J}0* /b-M/8b6 1-/-:- .3-0[b*L*,09†
                                                             2

   •ƒ |ha, bi| · .7-}6L? .63677*7-:36;;6\ .-/83-*,,-:- ,6 b*18-36N a 0 b =
   ]ƒ ha, bi > 0 ⇐⇒ {a, b} · .36bM +6[0/=
      ha, bi < 0 ⇐⇒ {a, b} · 7*bM +6[0/ =
          bi = 0 ⇐⇒ a||b =
       —;**8
   ƃ ha,      ;*/8- /7*L2J}69 O-3;276 L79 .7-}6L0 83*2:-7?,016 ABC
,6 .7-/1-/80†
                                 1 −→ −→         1    xB − xA yB − yA
              S4ABC =              |h AB, AC i| = mod                                            .
                                 2               2    xC − xA yC − yA

   ڃ ‡79 83*2:-7?,016 ABC ,6 .7-/1-/80† h −AB, AC i > 0 \*/70 -+N-L b*3K
                                             → −→
I0, A 7→ B 7→ C 7→ A .3*L/86b79*8 /-+-M Lb0|*,0* b-132: 83*2:-7?,016
b .-7-|08*7?,-; ,6.36b7*,00 ‚.3-80b )6/-b-M /83*710ƒ\0 h −AB,  AC i < 0 \
                                                            → −→
*/70 -+N-L b*3I0, A 7→ B 7→ C 7→ A 9b79*8/9 Lb0|*,0*; .- )6/-b-M
/83*71* =

                                                       ¸Ô

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