Аналитическая геометрия. Часть III. Многомерные пространства. Гиперповерхности второго порядка. Шурыгин В.В. - 29 стр.

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θ ∈ [0; π]
cos θ =
(x, y)
|x||y|
.
θ ∈ [0; π] (26)
x y E
n
E
n
A, B ∈
E
n
dist (A, B) = |
−→
AB|
A B
dist (A, B) = 0 ⇐⇒ A = B
dist (A, B) + dist (B, C) > dist (A, C).
|
−→
AC|
2
= |
−→
AB +
−−→
BC|
2
= (
−→
AB +
−−→
BC,
−→
AB+
−−→
BC) = |
−→
AB|
2
+2(
−→
AB,
−−→
BC)+|
−−→
BC|
2
6 |
−→
AB|
2
+2|
−→
AB||
−−→
BC|+ |
−−→
BC|
2
=
(|
−→
AB| + |
−−→
BC|)
2
.
E
n
E
n
α : E
n
→ E
n
GO(E
n
) E
n
GA(E
n
) E
n
α, β ∈ GO(E
n
) bα,
b
β ∈ O(E
n
)
[
α ◦ β = bα ◦
b
β ∈
O(E
n
)
E
n
{O; e
i
} α ∈ GO(E
n
)
y
i
= a
i
k
x
k
+ b
i
,
(a
i
k
)
.-¨7-:2 /2ˆ*/7B2*7 *D0,/7B*,,-* )0/6- θ ∈ [0; π] 751-*  )7-
                                cos θ =
                                          (x, y)
                                                 .                         •G·‹
                                          |x||y|
    VWXYZY[Y\]Y^ ïahog θ ∈ [0; π] w gnbglbjsbg gefmnmotmdgm `gfd}ogu (26) w
bjlckjmiht }xogd dmyn} kmvigfjda x a y mkvoangkj efghifjbhikj En €
    VWXYZY[Y\]Y^ ž}hi‚ E Ÿ mkvoangkg j``abbgm efghifjbhikg a A, B ∈
     ïahog
                         n
En €
                                             −→
                              dist (A, B) = |AB|
bjlckjmiht fjhhigtbamd dmyn} igsvjda A a B €
  ŽXYZ[—¡Y\]Y^ dist (A, B) = 0 ⇐⇒ A = B € 
  ŽXYZ[—¡Y\]Y^ Ødmmi dmhig homn}{qmm bmfjkmbhikgw bjlckjmdgm ðbmfjp
kmbhikgd ifm}xgo‚bavjñ¿
                   dist (A, B) + dist (B, C) > dist (A, C).
  £—™œ¤œšY[¥˜š›—^ -/.-6>E2*:/8 /B-”/7B5:0 /15683,-9- .3-0EB*D*,08 0
,*35B*,/7B-: -‰0 ì2,81-B/1-9- < :**:~ |−→      −→ −−→        −→
                                        AC|2 = |AB + BC|2 = (AB +
−−→ −→ −−→        −→  −→ −−→ −−→       −→      −→ −−→ −−→
BC, AB+BC) = |AB|2 +2(AB, BC)+|BC|2 6 |AB|2 +2|AB||BC|+|BC|2 =
  −→     −−→
(|AB| + |BC|)2 . 
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                                                                 n
   VWXYZY[Y\]Y^  kaymbamd mkvoangkj j``abbgxg efghifjbhikj E bjp
lckjmiht algdgf`ald mkvoangkcr j``abbcr efghifjbhik α : En → En €
                                                                          n

   @,-“*/7B- GO(E ) B/*C DB0“*,0” *B160D-B5 .3-/735,/7B5 E -+35E2
*7 932..2 -7,-/07*6>,- 1-:.-E0½00  8B68‡ˆ2‡/8 .-D932..-” B 932..*
                      n                                              n

               „„
GA(En ) B/*C 5 0,,AC DB0“*,0” .3-/735,/7B5 En < *”/7B07*6>,-  */60
                                                             ±‹‹ ◦ β = αb ◦ βb ∈
                    b, βb ∈ O(En ) 0  /6*D-B57*6>,- •/: < •G  α[
α, β ∈ GO(En )  7- α
O(En ) <
     /0/7*:* 1--3D0,57 B .3-/735,/7B* En  -.3*D*68*:-” -37-,-3:03-B5,
,A: 3*.*3-: {O; ei}  DB0“*,0* α ∈ GO(En) 0:**7 235B,*,08
                                y i = aik xk + bi ,
9D* (aik ) F -37-9-,56>,58 :5730½5 •/:< / < G·‹<
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