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(35 (42 (54 (65

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AC C A ( , , )

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a b c : V | a.(b c) |

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24

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: 25

﹁︭﹏ 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

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( ( ( ( 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

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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


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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