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ABC CA CB
C
π
3
ϕ
AA
1
BB
1
CC
1
{O; e
i
}
O = C e
1
=
−→
CA e
2
=
−−→
CB (g
ij
)
g
11
= 16 g
12
= g
21
= 12 g
22
= 36
C
A
1
B
B
1
A
(a, b) = 16a
1
b
1
+ 12a
1
b
2
+ 12a
2
b
1
+ 36a
2
b
2
.
m
a
=
−−→
AA
1
= {−1;
1
2
} m
b
=
−−→
BB
1
= {
1
2
; −1} m
c
=
−−→
CC
1
= {
1
2
;
1
2
}
m
2
a
= 16(−1)
2
+ 2 · 12(−1)(
1
2
) + 36(
1
2
)
2
= 13
m
2
b
= 16(
1
2
)
2
+ 2 · 12(
1
2
)(−1) + 36(−1)
2
= 28
(m
a
, m
b
) = 16(−1)(
1
2
) +12(−1)(−1)+12(
1
2
)(
1
2
) +
36(
1
2
)(−1) = −11 cos ϕ =
−11
√
13
√
28
m
2
c
= 16(
1
2
)
2
+ 2 ·12(
1
2
)
2
+ 36(
1
2
)
2
= 19
|m
c
| =
√
19
ABCD ψ
BB
1
CC
1
ABC ACD
•
•
•
•
•
•
B
C
A
D
C
1
B
1
ABCD 1
{O; e
i
} O = A
e
1
=
−→
AB e
2
=
−→
AC e
3
=
−−→
AD
(g
ij
)
g
11
= g
22
= g
33
= 1 g
12
=
g
21
= g
13
= g
31
= g
23
= g
32
=
1
2
(a, b) = a
1
b
1
+
1
2
a
1
b
2
+
1
2
a
1
b
3
+
1
2
a
2
b
1
+a
2
b
2
+
1
2
a
2
b
3
+
1
2
a
3
b
1
+
1
2
a
3
b
2
+ a
3
b
3
.
u =
−−→
BB
1
= {−1;
1
2
; 0} v =
−−→
CC
1
= {0; −1;
1
2
} (u, v) =
1
2
(−1)(−1) +
1
2
(−1)(
1
2
) +
1
2
(−1) +
1
2
(
1
2
)(
1
2
) = −
1
8
|u| = |v| =
√
3
2
cos ψ = −
1
12
ðÍwÍñÍ rÊZ { 83*2:-7?,01* ABC L70, /8-3-, CA 0 CB 36b,\ /--8K
b*8/8b*,,-\ Ú 0 _\ 6 2:-7 .30 b*3I0,* C 36b*, = 6M80 2:-7 ϕ ;*|L2 π
;*L06,6;0 AA 0 BB = ] 6M80 L70,2 ;*L06, CC = 3
àxÿxXWxZ ;= 30/2,-1 ]ò= {b*L*; ,6 .7-/1-/80 3*.*3 {O; e } \ .-76:69
1 1 1
−−→ { )
O = C \ e = CA \ e = CB = 0/70; ~7*;*,8 ;6830 (g ) L79 bK
i
−→
+36,,-:- +6[0/6 g = 16 \ g = g = 12 \ g = 36 =
1 2 ij
¹-3;276 ã .30,0;6*8 b0L
11 12 21 22
C
(a, b) = 16a1 b1 + 12a1 b2 + 12a2 b1 + 36a2 b2 .
B1 {*18-3 ;*L06, 0;*J8 b ~8-; +6[0/* /7*L2JK
A }0* 1--3L0,68 m = −AA
1 −→
= {−1; } \ m =
1
BB = { ; −1} \ m = CC = { ; } = ;**;
a 1 2 b
−−→ −−→ 1 1 1
m = 16(−1) + 2 · 12(−1)( ) + 36( ) = 13 \
1 2 c 1 2 2
A 2 2 1 1 2
m = 16( ) + 2 · 12( )(−1) + 36(−1) = 28 \
a 2 2
1 2 1
Ç0/= ]ò=
2 2
b 2 2
B
1 1 1
(m , m ) = 16(−1)( )+12(−1)(−1)+12( )( )+
36( )(−1) = −11 = 7*L-b68*7?,-\ cos ϕ =
a b
=
2 2 2
1 √ −11
5,67-:0),-\ m = 16( ) + 2 · 12( ) + 36( ) = 19 \ .-~8-;2 |m | = √19 =
√
2 13 28
2 1 2 1 2 1 2
ðÍwÍñÍ rZ { .36b07?,-; 8*836~L3* ABCD ,6M80 2:-7 ψ ;*|L2 ;*L0K
c 2 2 2 c
6,6;0 BB 0 CC :36,*M ABC 0 ACD =
1 1
àxÿxXWxZ ;= 30/2,-1 ]_= Â2L*; /)0K
A
868?\ )8- L70,6 3*+36 36//;6830b6*;-:-
8*836~L36 ABCD 36b,6 1 = {b*L*; b .3-K
•
C 1
/836,/8b* 3*.*3 {O; e } \ .-76:69 O = A \
•
e = AB \ e = AC = e = AD = 7*;*,K
B 1 i
• −→ •−→ −−→
D
8 ;6830 (g ) L79 b+36,,-:- +6[0/61 2 3
• 0;*J8 b0L g = g = g = 1 \ g = ij
g = g = g = g = g = = ¹-3K
11 22 33 12
B •
1
C
Ç0/= ]_= ;276 a .30,0;6*8 b0L (a, b) = a b +
21 13 31 23 32 2
1 1
{
1 1 2 1 1 3 1 2 1 2 2 1 2 3 1 3 1
a b + a b + a b +a b + a b + a b +
a b + a b . *18-3 ;*L06, 0;*J8 b ~8-; +6[0/* /7*L2J}0* 1--3L0K
2 2 2 2 2
1 3 2 3 3
,68 u = −BB
2
−→
= {−1; ; 0} \ v = CC = {0; −1; } = -:L6 (u, v) =
−−→ 1 1
(−1)(−1) + (−1)( ) + (−1) + ( )( ) = − = 61 161 |u| = |v| = \
1 2 1 2 √
1 1 1 1 1 1 1 1 3
8- cos ψ = − =
2
1
2
12
2 2 2 2 2 8 2
Á
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