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a b (x x , y y ,z z )
AB B A
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oz
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11 ®\a. aópG [... ] .
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22 ·°Ãa ®\a ] ... [ .
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a .b | a || b | cos
V (a ,b,c) V a i b j ck
a .b xx yy zz
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6 a b c a b c o .a .b a .c b .c 4 24
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12
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7 a b c . a b 4 24
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: 13
a .bcos | a || b |
11 a (m , , )2 1 | b | 41 a b a b m
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a b a b (a b).(a b) | a | | b | | a | | b | m m2 2 2 2 20 0 4 1 41 6
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c o0 0
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14
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16 , ,33 4 4
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17 a ( , , )0 3 6 b ( , , )2 1 2 3 24
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2 2 290 02 1 0 3 2 4521 1 4 1 4 2 3
", `0\a? E¼O `0\a? · Ã a (x , y,z)
: :ae ae (cos ,cos ,cos ) .
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: 17
| a .b | | a || b |
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| a || b |
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20
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(35 (42 (54 (65
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a b c : V | a.(b c) |
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: 21
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2 2 2 3 2 !! :
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40 a i j2 b j k3 c i k4 . a b 1 32
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22
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: 23
48 m a ( , ,m)1 2 V ( , , )1 0 1 2 V ( , , )2 2 3 1
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a .(b c) b.(c a) c.(a b)
49 a b c 9 128
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: (a c).b b.(a c) a .(c b)
50 ab c ( a b).((b c) (c a))2 6 33
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o( a b).((b c) (c a)) ( a b).(b c b a c c c a)2 2
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4 2 34 1 2
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﹁︭﹏ 11دوره ایآز﹝﹢ن ︨﹣ا﹐ت
1 - | a | 2 | b | 4a b c o2 a .b b .c a .c 4 24 87
( ( (10 (20 2 - ( , , )1 1 2 ( , , )2 0 1 92 87
(56 (142 (14 (7
3 - a b c o3 2 7 33 70
(c a3 (c b2 (a b6 (b c3 4 - | a | | b | | a b | 6 | a b | 2 3 a b 81
(3 3 ( ( (6 3 5 - b c a b2c a
abc 6 33 1 32 90 84
( (/7 5 ( ( 6 - a.b 2 a.c 3 b.c 4 (a b) c 10 33 92
(a b4 3 (b a3 4 (b a3 2 (b a3 2 7 - a i j k3 2 b i j k2 . a b8 7 a b11 9
(( , , )3 1 6 (( , , )6 2 10 2 32 88 (( , , )2 6 12 (( , , )3 1 5
8 - a ( , , )3 1 1 b ( , , )1 0 1 c a b 3 ab c 31 89
(2 3 (4 3 (3 2 (12
1 2 3 411 2 3 42
1 2 3 431 2 3 44
1 2 3 451 2 3 46
1 2 3 471 2 3 48
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26
﹁︭﹏ 21دوره ایآز﹝﹢ن︨﹣ا﹐ت
1 - a i jb j k c i k
( (2 (3 ( 84 2 - a i j k2 .
aa j 86
(152 (30
2 (15 (30
3 - a ox oy60 a b b c c a i j2 4 abc 9 33 91
( ( ( ( 4 - a i k3 b i c a.c 90
(- ( ( (- 5 - a b 90 0 3/
a b2 81
(415 (3
5 (415 (3
5
6 - abc 2 4 b c 150 a b c | a .(b ( a c)) |2 90
( (8 3 ( ( 7 - a ( , , )2 3 1 b (m , , )2 4 m ( a b).(a b)2 3
88
(m 2 (m 2 (m (m 8 - a ( , , )2 0 1 b ( , , )1 2 0 c a c ,2( )
c 90
(( , , )1 1 2 16 (( , , )1 1 2 16 (( , , )1 1 2 16 (( , , )1 1 2 16
1 2 3 411 2 3 42
1 2 3 431 2 3 44
1 2 3 451 2 3 46
1 2 3 471 2 3 48
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: 27
11دوره ایآز﹝﹢ن︎︀︨ـ ﹏︭﹁
1 3 a .b b.c a .c | a b c |2 :
a b c o a b c c | a b c | | c | | a b c | | c |2 22
| a | | b | | c |2 2 2
x
(a .b b.c a .c) | c |22 (x) x4 16 2 0 10
2 2 a bS | a b |
a ba b :
o o S
| (a b) (a b) | | a a a b b a b b | | a b a b | | a b | | a b | S2 2 2
.
S | ( , , ) ( , , ) | | ( , , ) | S 142 1 1 2 2 0 1 1 3 2 1 9 4 14 2
3 4 x y z o x y y z z x :
x y za b c o a ( b) ( b) c c ( a) a b b c c a3 2 3 2 2 3 6 2 3
4 1 a b : S | a b |1
2 | a || b | sin.
|a| |b|| a b | | a b | (| a | | b | ) (| a | | a | ) | a | | a | | b |2 2 2 2 2 2 22 36 12 2 12 12''''''" | a b | | a b | a .b a .b a .b2 2 4 36 12 4 6 a .b | a || b | cos cos cos 16 12 12 602
S | a b | | a || b | sin sin1 1 1 12 12 60 3 32 2 2
5 1 o
(b c).(( a b) (c a)) (b c).( a c a a b c b a)2 15 2 2 15
b.(a c)
b.( a c) b.(b c) b.(b a) c.( a c) c.(b c) c.(b a) b.(a c) b.(a c) b.(a c)0 0 0 0
2 2 15 2 3 15 b.(a c) 5
6 1 (a b) c c (a b) c (b a) (a.c)b a (b.c) b a3 4
7 2 a b b a ab .
( a b) ( a b)8 7 11 9 a b . . ( )a( , , )
a b ( , , ) ( , , )b( , , )
23 1 2 3 1 5 6 2 102 1 1 ''''''"
8 3 ca b . ab :
a ( , , )a b ( , , ) S | a b |
b ( , , )3 1 1 1 2 1 61 0 1 ''"
V Sh 6 3 18 3 2
c,b ,a
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28
1دوره ایآز﹝﹢ن︎︀︨ـ ﹏︭﹁ 2
1 1 ijk :
V1 0 00 1 0 10 0 1
2 2 a j ( , , )
(a j) a ( , , ) S | a (a j) |a ( , , )
1 0 2 1 1 302 5 1 4 25 12 2 22 1 1 ''"a ( , , )j ( , , )
2 1 10 1 0
3 4 a :
a .(a b) a .(b c) a .(c a) a .i a . j a .(b c) | a || i | cos | a || j | cos0 0
2 4 2 4
a .(b c) V | a .(b c) |1 12 2 4 2 62 2
4 3 | a | | c | a c . a b :
a.bcos | a || b |3 3 1502 1 2
a.c | a || c | cos 160 2 2 22
5 2 :
S | a b | | a || b | sin sin cos1 10 3 0 6 0 82 2/ / /
| a b | | a | | b | | a || b | cos | a b |2 2 2 8 9 32 4 4 4 1 4 210 5 5( )
6 3 :
o| a .(b ( b c)) | | a .( b b b c) | | a .(b c) | | a .(b c)|2 2
| a || b || c | sin cos 1150 0 2 2 4 1 82
7 4 a b a b2 3 m m .
8 3 a b , ( , , )1 11 1 2 12 2 2,
( )d
d ce ed ce ( , , ) e ( , , )1 11 2 1 1 2 16 6''''''''''"
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