Аналитическая геометрия. Часть I. Шурыгин В.В. - 32 стр.

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pr
a
a b V
3
ϕ
a b
(a, b) = |a||b|cos ϕ.
a ⊥ b ⇐⇒ (a, b) = 0 0 ⊥ b
b ∈ V
3
ϕ a b
0 6 ϕ 6 π
(a, b) > 0 ⇐⇒ ϕ <
π
2
(a, b) < 0 ⇐⇒ ϕ >
π
2
(a a) = |a|
2
pr
a
(b) =
(a,b)
|a|
1
◦
(a, b) = (b, a)
2
◦
(λa + µb, c) = λ(a, c) + µ(b, c)
3
◦
(c, λa + µb) = λ(c, a) + µ(c, b)
4
◦
a 6= 0 =⇒ (a, a) > 0
1
◦
4
◦
3
◦
1
◦
2
◦
2
◦
c = 0 2
◦
c 6= 0
(λa + µb, c) = |c|pr
c
(λa + µb) = λ|c|pr
c
(a) + µ|c|pr
c
(b) = λ(a, c) + µ(b, c)
V W
ϕ : V × V → W
ϕ(λa + µb, c) = λϕ(a, c) + µϕ(b, c)
ϕ(c, λa + µb) = λϕ(c, a) + µϕ(c, b)
+2L*; 0/.-7?[-b68? 861|* -+-[,6)*,0* pr =                      a

Z± sy͙´TXux ³TuWêtxwxXWxZ
ϳTxwx™xXWxZ ›œžŸ a ¥ b ”Ž‰žŠ‘ Œ¢ V ¥  ϕ œ©Š‹ ’Ž»ªœ Œ’Œ§
ú‰‹£‘’ ¡ŠŒ¢”ŽªŽŒŽ’ ”Ž‰žŠŠ” a Œ b ¢‘”Žž£ ¤Œ‹Š
                                                                   3




                                            (a, b) = |a||b| cos ϕ.
  xuµxzTWñx¶yWx ¶tu̶ztÍ ¶y͙´TXuVu ³TuWêtxwxXW´Z
   ­7*L2J}0* /b-M/8b6 /16793,-:- .3-0[b*L*,09 b*18-3-b b8*16J8 ,*.-K
/3*L/8b*,,- 0[ -.3*L*7*,09†
   •ƒ a ⊥ b ⇐⇒ (a, b) = 0 ‚.- -.3*L*7*,0J /)086*8/9\ )8- 0 ⊥ b L79
7J+-:- b ∈ V ƒ=
   ]ƒ H3*L.-76:69\ )8- 2:-7 ϕ ;*|L2 a 0 b 2L-b7*8b-39*8 /--8,-I*,0J
                3


0 6 ϕ 6 π \ 0;**;†
   (a, b) > 0 ⇐⇒ ϕ < \              π

   (a, b) < 0 ⇐⇒ ϕ > =
                                    2

   ƃ (a \ a) = |a| =
                                    π
                                    2
                      2
   ڃ pr (b) = =
        a
                    (a,b)
                     |a|

     ™VxvTÍWñx¶yWx ¶tu̶ztÍ ¶y͙´TXuVu ³TuWêtxwxXW´Z
   1 = (a, b) = (b, a) =
    ◦

   2 = (λa + µb, c) = λ(a, c) + µ(b, c) =
    ◦

   3 = (c, λa + µb) = λ(c, a) + µ(c, b) =
    ◦

   4 = a 6= 0 =⇒ (a, a) > 0 =
    ◦
   éuyÍêÍzx™ë¶ztuZ 1 0 4 b8*16J8 ,*.-/3*L/8b*,,- 0[ -.3*L*7*,09 = 3
/7*L2*8 0[ 1 0 2 = ‡-16|*; 2 =
                                ◦       ◦                                   ◦


   ¼/70 c = 0 \ 8- 2 b.-7,9*8/9 -)*b0L,; -+36[-;= ¼/70 c 6= 0 \ 8-
            ◦         ◦                       ◦

                                ◦

(λa + µb, c) = |c|pr (λa + µb) = λ|c|pr (a) + µ|c|pr (b) = λ(a, c) + µ(b, c) =
   ϳTxwx™xXWxZ ›œžŸ V Œ W ”Ž‰žŠ‘Ž ¡Šžž”§ ¿žŠ¬»Žº
                            c                              c           c


ŒŽ ϕ : V × V → W ¢‘”Žž£ ¬Œ‹ŒŽ¦‘’¥ Ž‹Œ
   ϕ(λa + µb, c) = λϕ(a, c) + µϕ(b, c) \
   ϕ(c, λa + µb) = λϕ(c, a) + µϕ(c, b) =

                                                      –ï

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