Аналитическая геометрия. Часть I. Шурыгин В.В. - 35 стр.

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V
3
(a, b) = g
11
a
1
b
1
+ g
12
a
1
b
2
+ g
13
a
1
b
3
+ g
21
a
2
b
1
+ g
22
a
2
b
2
+
+ g
23
a
2
b
3
+ g
31
a
3
b
1
+ g
32
a
3
b
2
+ g
33
a
3
b
3
.
V
2
(a, b) = g
11
a
1
b
1
+ g
12
a
1
b
2
+ g
21
a
2
b
1
+ g
22
a
2
b
2
.
{e
i
} |e
i
| =
1 i, j = 1, 2, 3, e
i
⊥ e
j
i 6= j
(g
ij
) g
ij
=
δ
ij
V
3
V
2
(a, b) = a
1
b
1
+ a
2
b
2
+ a
3
b
3
,
(a, b) = a
1
b
1
+ a
2
b
2
.
ABC AA
1
BB
1
CB CA
• •
•
• •
•
A B
B
1
A
1
C
O
|
−−→
AA
1
| = |
−−→
BB
1
| = 3
O
{O, e
i
}
e
1
=
−−→
OA
1
e
2
=
−−→
OB
1
−−→
CB = 2
−−→
A
1
B = 2(
−−→
A
1
O +
−−→
OB) = 2{−1; −2} =
{−2; −4}
−→
CA = 2
−−→
B
1
A = 2(
−−→
B
1
O+
−→
OA) = 2{−2; −1} =
{−4; −2}
cos ∠C =
16
(
√
20)(
√
20)
=
4
5
/-/86b7*,,2J 0[ /16793,N .3-0[b*L*,0M ‚•ãƒ b*18-3-b +6[0/6= ‡79 .3-K
/836,/8b6 V b 36[b*3,28-; b0L* O-3;276 ‚•_ƒ b:79L08 /7*L2J}0; -+K
36[-;†         3



     (a, b) = g11 a1 b1 + g12 a1 b2 + g13 a1 b3 + g21 a2 b1 + g22 a2 b2 +
                                                                                                                 ‚•aƒ
                                             + g23 a2 b3 + g31 a3 b1 + g32 a3 b2 + g33 a3 b3 .
{ /72)6* .7-/1-/80 V \ /--8b*8/8b*,,-\ 0;**;†
                                2

                       (a, b) = g11 a1 b1 + g12 a1 b2 + g21 a2 b1 + g22 a2 b2 .                                  ‚•`ƒ
    ϳTxwx™xXWxZ ü¢Œ {e } ¢‘”Žž£ ŠžŠŠ’ŒŠ”‘’¥ Ž‹Œ |e | =
1 ¥ i, j = 1, 2, 3, Œ e ⊥ e ¡Œ i 6= j =
                                        i                                                                        i


    { /72)6* -38-,-3;03-b6,,-:- +6[0/6 ;6830 6 (g ) · *L0,0),69\ g =
                           i        j



δ \ 0 /16793,* .3-0[b*L*,09 b*18-3-b b V 0 V ,6N-L98/9\ /--8b*8/8b*,K
                                                                                    ij                           ij


,-\ .- /7*L2J}0; O-3;276;†
ij                                                                  3       2




                                    (a, b) = a1 b1 + a2 b2 + a3 b3 ,

                                            (a, b) = a1 b1 + a2 b2 .

Z ˜TWµxTUZ
H30b*L*; .30;*3\ 077J/83032J}0* .30;*,*,0* O-3;27 ‚•_ƒ=
   ðÍwÍñÍ r±Z { 36b,-+*L3*,,-; 83*2:-7?,01* ABC ;*L06, AA 0 BB \
.3-b*L*,,* 1 +-1-b; ‚36b,;ƒ /8-3-,6; CB 0 CA \ .*3*/*16J8/9 .-L                                       1            1


.39;; 2:7-;= €6M80 2:7 ~8-:- 83*2:-7?,016=
   àxÿxXWxZ ­;= 30/2,-1 ]Ú = H-/1-7?12 b/* 83*K
2:-7?,010\2L-b7*8b-39J}0* 2/7-b09; [6L6)0\.-K              C                                         •



L-+,\ +2L*; /)0868?\ )8- |−AA
                             −→
                                | = |BB | = 3 =
                                       −−→
H2/8? O · 8-)16 .*3*/*)*,09 ;*L06,= {b*L*; ,6     1
                                                    B        A
                                                                1
                                                                                             1               1
.7-/1-/80 -38-,-3;03-b6,,M 3*.*3 {O, e } \ bK
                                                                                                 •       •

                                                                                                     •


+36b [6 b*18-3 +6[0/6 e = −OA−→ \
                                   e = OB = „-K
                                         −−→             O
                                                                    i


:L6 −CB
     −→     −−→      −−→ −−→
                                        1
        = 2A B = 2(A O + OB) = 2{−1; −2} =
                                                   1   2                1
                                                                                         •                   •


         \ −→
                   1
                 −−→      −−→ −→
                                    1
{−2; −4} CA = 2B A = 2(B O+OA) = 2{−2; −1} =
                                                   A
                                                       0/ ]Ú
                                                      Ç = =
                                                               B

{−4; −2} = ˆ8/JL6 ,6N-L0;   1
                           cos ∠C =
                                             1
                                             = =            √ 16√               4
                                                                                5
                                                           ( 20)( 20)
                                    –¾

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