ﺔﻓﺪﮭﺘﺴﻤﻟا تاءﺎﻔﻜﻟا 7 7

(1)

تاءﺎﻔﻜﻟا ﺔﻓﺪﮭﺘﺴﻤﻟا

ﻥﻴﺘﻁﻘﻨ ﺢﺠﺭﻤ ﺀﺎﺸﻨﺇ

.

ﻁﻘﻨ ﺙﻼﺜ ﺢﺠﺭﻤ ﺀﺎﺸﻨﺇ .

ﺢﺠﺭﻤﻟﺍ ﺕﺎﻴﺜﺍﺩﺤﺇ ﺏﺎﺴﺤ .

ﻁﻘﻨ ﺔﻴﻤﺎﻘﺘﺴﺍ ﺕﺎﺒﺜﻹ ﺢﺠﺭﻤﻟﺍ لﺎﻤﻌﺘﺴﺍ ﺕﺎﻤﻴﻘﺘﺴﻤ ﻲﻗﻼﺘ ﻭﺃ .

Ã

ﻊﯿﻤﺠﺘﻟا ﺔﯿﺻﺎﺧ ﻞﺼﻔﻟا اﺬھ ﻲﻓ ﮫﯿﻓ ﻢﻜﺤﺘﻟا ﻲﻐﺒﻨﯾ ﺎﻣ ﻢھأ

Ã

ﺔﻋﻮﻨﺘﻣ تﻼﻜﺸﻣ ﻞﺣ ﻲﻓ ﺔﻟﺎﻌﻓ ةادأ ﺢﺟﺮﻤﻟا ﺮﺒﺘﻌﯾ ﻲﻗﻼﺗ تﺎﺒﺛإ و ﻂﻘﻧ ﺔﻋﻮﻤﺠﻣ ﻦﯿﯿﻌﺘﻛ )

ةﺪﺣاو ﺔﻄﻘﻧ ﻲﻓ تﺎﻤﯿﻘﺘﺴﻣ

Ã ﻠﻌﺘﻤﻟا ﻰﻠﻋ ﻢ

ﺔﻗﻼﻌﻟا ﺔﻤﺟﺮﺗ ﯿﻋﺎﻌﺷﻹا

ﺔ ﺲﻜﻌﻟا و ﺢﺟﺮﻤﻟا ﺎﮭﻘﻘﺤﯾ ﻲﺘﻟا

Ã

ﺔﻠﺴﻠﺳ لﺪﻌﻣ و ﺢﺟﺮﻤﻟا ﻦﯿﺑ ﺔﻗﻼﻌﻟا ﻢﻠﻌﺘﻤﻟا ﻆﺣﻼﯾ تﺎﻘﯿﺒﻄﺘﻟا ﻲﻓ ﺔﻟﺎﻄﻌﻟا ﺰﻛﺮﻣ و ﺔﯿﺋﺎﺼﺣإ

ﺔﯿﺋﺎﯾﺰﯿﻔﻟا

7

(2)

لوﻷا طﺎﺸﻨﻟا :

فﺪﮭﻟا ﻦﯿﺘﻄﻘﻧ ﺢﺟﺮﻣ مﻮﮭﻔﻣ جاردإ :

1 ﺢﯿﺤﺼﺗ ( ﺔﻤﯿﻗ ﺐﺴﺣأ :

mB

ﺔﻟﻻﺪﺑ و GA

ضﻮﻋ GB GAuuur

و GBuuur باﻮﺠﻟا و

B 6 m GA

= GB

2 ( 3 * GAuuur= −7GBuuur *

ﻊﻀﻧ : BG =BA+AG uuur uuur uuuur *

AG = 6 Cm

3 * ( ﺬﺧﺄﻧ

B 2 A

m = m *

ﺬﺧﺄﻧ

B 5 A

m = m

ﻲﻧﺎﺜﻟا طﺎﺸﻨﻟا :

فﺪﮭﻟا ﻂﻘﻧ ثﻼﺛ ﺢﺟﺮﻣ ءﺎﺸﻧإ :

1 1 (

AI =3AB uuur uuur و

2 BJ = BC uuur uuur

2 ﺔﻗﻼﻌﻟا ﻲﻓ ( 2AGuuuur uuur+GB−2GCuuur

ﻊﻀﻧ GAuuur uur uur=GI +IA و

GBuuur uur uur=GI +IB ﻊﻣ

G , I , C ةﺪﺣاو ﺔﻣﺎﻘﺘﺳا ﻰﻠﻋ

3 ﻊﻀﻧ ﺎﮭﺴﻔﻧ ﺔﻗﻼﻌﻟا ﻲﻓ ( GBuuur uuur uur=GJ +JB

و GCuuur uuur uuur=GJ +JC ﻊﻣ

G , J , A ةﺪﺣاو ﺔﻣﺎﻘﺘﺳا ﻰﻠﻋ

4 ( ﻊﻃﺎﻘﺗ ﺔﻄﻘﻧ G )

AJ و ( ) GI ( 6 ﻊﻀﻧ ﺔﻘﺑﺎﺴﻟا ﺔﻗﻼﻌﻟا ﻲﻓ ( GAuuur uuur uuur=GC +CA

ﺚﻟﺎﺜﻟا طﺎﺸﻨﻟا :

فﺪﮭﻟا ﻦﯿﺘﻄﻘﻧ ﺞﺣﺮﻣ ﻦﯿﯿﻌﺗ :

1 ( 2 GAuuur= − GBuuur يأ

2 AG = 3AB uuuur uuur

2 3 (

AG =5AB uuuur uuur

3 ( ﻒﺼﺘﻨﻣ G ]

AB [ 4 ( m = 4 Kg

ﻊﺑاﺮﻟا طﺎﺸﻨﻟا :

فﺪﮭﻟا

ﻦﯿﯿﻌﺘﻟ ﻊﯿﻤﺠﺘﻟا ﺔﯿﺻﺎﺧ لﺎﻤﻌﺘﺳا : ﺔﻠﻤﺟ ﺢﺟﺮﻣ

.

1 ( 10, 77 m; 2

1 11, 07 ( m ; ،

2 9, 76 m ;

3 13,83 m ;

3 ﺮﺴﻜﻟا مﺎﻘﻣ ﺢﯿﺤﺼﺗ ( ﺲﯿﻟ و 29

28 .

1 2 3

13 13 3

10, 77 29

m + m + m

;

ﺔﮭﺟﻮﻣ لﺎﻤﻋأ 1

: فﺪﮭﻟا ﻦﯿﯿﻌﺗ :

ﺢﺟﺮﻤﻟا لﺎﻤﻌﺘﺳﺎﺑ ﻂﻘﻧ ﺔﻋﻮﻤﺠﻣ

1 ( ﺎھﺰﻛﺮﻣ ةﺮﺋاد C ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ G

} ) 3 ( ،C ) 2 - ( ،B ) 1 ( A ﺎھﺮﻄﻗ ﻒﺼﻧ و { 3

} ) 3 - ( ،C ) 1 ( ،B ) 2 - ( A ﺎھﺮﻄﻗ ﻒﺼﻧ و { 5

ﻦﯿﺘﻄﻘﻨﻟا ﻞﻤﺸﺗ ﻲھ و 4 و A

C

} ) 2 ( ،C ) 1 - ( ،B ) 1 ( A ﺎھﺮﻄﻗ ﻒﺼﻧ و { 3

ﻞﻤﺸﺗ ﻲھ و 2 A

4 ﺔﻌﻄﻘﻟا رﻮﺤﻣ ﻲھ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ( ]

GP ﺚﯿﺣ [ ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ G }

) 2 ( ،C ) 1 ( ،B ) 1 ( A {

و ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ P }

) 1 ( ،C ) 1 ( A {

5 * ( ةاوﺎﺴﻤﻟا ﻖﻘﺤﺗ B موﺪﻌﻣ تﻼﻣﺎﻌﻤﻟا عﻮﻤﺠﻣ *

* 3 BAuuur uuur+BC = α

) * Γ ﺎھﺰﻛﺮﻣ ةﺮﺋاد ( ﺎھﺮﻄﻗ ﻒﺼﻧ و G

3 2 α * ءﺎﺸﻧﻺﻟ

Γ ) ﻞﻤﺸﺗ ( B

:

فﺪﮭﻟا تﺎﻤﯿﻘﺘﺴﻣ ﻲﻗﻼﺗ تﺎﺒﺛﻹ ﺢﺟﺮﻤﻟا لﺎﻤﻌﺘﺳا :

1 ءﺎﺸﻧﻺﻟ ( : 3 *

AI =2AB uuur uuur *

3 AJ = AC uuur uuuur

تﺎﻤﯿﻘﺘﺴﻤﻟا نأ وﺪﺒﯾ )

CI ، ( ) BJ و ( ) AK ةﺪﺣاو ﺔﻄﻘﻧ ﻲﻓ ﻊﻃﺎﻘﺘﺗ (

2 ( ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ I }

) 3 - ( ،B ) 1 ( A ، { ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ J }

) 3 - ( ،C ) 2 ( A ، { ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ K }

) 1 ( ،C ) 2 ( B {

(3)

3 ﺮﺒﺘﻌﻧ ( ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ G }

) 3 - ( ،C ) 6 - ( ،B ) 2 ( A ، { موﺪﻌﻣ ﺮﯿﻏ تﻼﻣﺎﻌﻤﻟا عﻮﻤﺠﻣ نﻷ دﻮﺟﻮﻣ G

ﻊﻤﺠﻟا ﺔﯿﺻﺎﺧ لﺎﻤﻌﺘﺳﺎﺑ :

ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ G }

) 3 - ( ،C ) 4 - ( I ﮫﻨﻣ و { G∈( IC)

) 6 - ( ،B ) 1 - ( J ﮫﻨﻣ و { G∈( BJ)

) 2 ( ،A ) 9 - ( K ﮫﻨﻣ و { G∈( AK)

: فﺪﮭﻟا رﻻوأ ﻢﯿﻘﺘﺴﻣ ﻰﻠﻋ فﺮﻌﺘﻟا :

2 ( 2 AH =AO+OH =AO+OA+ OA′

uuuur uuuur uuuur uuuur uuur uuuur ﻲﻟﺎﺘﻟﺎﺑ و

) AH ﻰﻠﻋ يدﻮﻤﻋ ( )

BC (

نﻷ OA′)

ﻰﻠﻋ يدﻮﻤﻋ ( )

BC (

3 ( ا ﻰﻘﺘﻠﻣ H ﺚﻠﺜﻤﻟا ﻲﻓ تﺎﻋﺎﻔﺗرﻹ ABC

4 ﺢﯿﺤﺼﺗ ( : ﻦﯿﺘﻄﻘﻨﻟا ﺢﺟﺮﻣ O و H

ﻦﯿﻠﻣﺎﻌﻤﻟﺎﺑ ﻦﯿﺘﻘﻓﺮﻣ G )

1 - و ( ) 3 ، ﺐﯿﺗﺮﺘﻟا ﻰﻠﻋ ( 3

OHuuuur= OGuuur

5 ( ، O ، G ، ةﺪﺣاو ﺔﻣﺎﻘﺘﺳا ﻰﻠﻋ H

[ ]

G∈ OH

ﺄﻄﺧ مأ ﺢﯿﺤﺻأ

(1 ﻰﻟإ (10

[ ]

¹

[ ]

²

[ ]

³

[ ]

⁴

[ ]

⁵

[ ]

⁶

[ ]

⁷

[ ]

⁸

[ ]

⁹

[ ]

¹⁰

ﺢﯿﺤﺻ ﺄﻄﺧ

ﺄﻄﺧ ﺢﯿﺤﺻ

ﺄﻄﺧ ﺄﻄﺧ

ﺄﻄﺧ ﺢﯿﺤﺻ

ﺢﯿﺤﺻ ﺢﯿﺤﺻ

تارﺎﯿﺘﺧﻹا ةدﺪﻌﺘﻣ ﺔﻠﺌﺳأ

(11

ﻰﻟإ (17

[ ]

¹¹

[ ]

¹²

[ ]

¹³

[ ]

¹⁴

[ ]

¹⁵

[ ]

¹⁶

[ ]

¹⁷

(3 ﺪﺟﻮﯾ ﻻ

(2

(3

(2

ﻦﯿﺘﻄﻘﻧ ﺢﺟﺮﻣ

:18

(1 ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ ﻮھ

( )( )

{

^A^{, 2} ^B ^{, 3}

}

(2

ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ ﻮھ

( ) ( )

{

^A^,1 ^B ^{, 2}

}

(3

ﻞﺑﺎﻘﻤﻟا ﻞﻜﺸﻟا ﻲﻓ

(4

ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ ﻮھ

( ) ( )

{

^A^{, 3} ^B ^{, 2}

}

: 19

ﻂﻘﻨﻟا ﺄﺸﻨﺗ

5

,

4

,

3

,

2

,

1

G G G G G

ﻦﯾﺮﻤﺘﻟا ﺔﻘﯾﺮﻃ ﺲﻔﻨﺑ 18

:20

21

22

ﺔﻟﺎﺤﻟا (1 ﺔﻟﺎﺤﻟا

(2 ﺔﻟﺎﺤﻟا

(5

{ ⁽

^A^{, 3}

^{) (}

^B^,⁻²

⁾ }

{ ⁽

^A^{, 1}

^{) (}

^B^,⁻³

⁾ }

{ ⁽

^A^{, 2}

^{) (}

^B ^{, 1}

⁾ }

{ (

^A^{, 1}

) (

^B ^{, 2}

) }

( )( )

{

^A^{, 2} ^B ^{, 1}⁻

}

ﺔﻟﺎﺤﻟا (1

(

^{α β}^,

) ( )

⁼ ^{2 , 1}

(

^{α β}^,

) (

⁼ ^{5 ,}⁻⁷

)

(

^{α β}^,

) (

⁼ ^{3 ,}⁻²

)

(

^{α β}^,

) (

⁼ ^{2 , 1}⁻

)

ﺔﻟﺎﺤﻟا 1

( )( )

{

^,1 ^,1

}

G = A B

ﺔﻟﺎﺤﻟا 2

( )( )

{

^{, 5} ^{, 3}

}

G = A B −

ﺔﻟﺎﺤﻟا 3

( )( )

{

^{, 5} ^{, 6}

}

G = A B −

ﺔﻟﺎﺤﻟا 4

( )( )

{

^{, 7} ^{, 3}

}

G = A − B

ﺔﻟﺎﺤﻟا 5

( )( )

{

^,1 ^{, 6}

}

G = A B −

ﺔﻟﺎﺤﻟا 6 ﺔﻠﻘﺜﻣ ﺔﻠﻤﺠﻟ ﺎﺤﺟﺮﻣ ﺖﺴﯿﻟ G

(4)

ﻦﯾﺮﻤﺘﻟا : 23

( ) ( )

{

^{, 1} ^{, 3}

}

A= C B − ،

( )( )

{

^{, 2} ^{, 1}

}

B = A C

( ) ( )

{

^{, 1} ^{, 4}

}

A= C B − ،

( )( )

{

^{, 3} ^, ⁴

}

C = A B −

( )( )

{

^{, 1} ^{, 1}

}

B = A C

( )( )

{

^{, 1} ^{, 2}

}

C = A − B

ﻦﯾﺮﻤﺘﻟا :26

ﺔﯿﻋﺎﻌﺸﻟا ةاوﺎﺴﻤﻟا مﺪﺨﺘﺴﻧ 0

CA CB αuuur+βuuur r= لﺎﺷ ﺔﻗﻼﻋو

) CBuuur uuur uuur=CA+AB (

ﻦﯾﺮﻤﺘﻟا 7 : 2 G1

ﺮﯿﻈﻧ ﻮھ ﻰﻟإ ﺔﺒﺴﻨﻟﺎﺑ A

َو B G2

ﺮﯿﻈﻧ ﻮھ ﻰﻟإ ﺔﺒﺴﻨﻟﺎﺑ C

B

(1 وﺎﺴﻤﻟا ماﺪﺨﺳﺎﺑ ﺊﺸﻨﻧ ﻦﯿﺘﯿﻋﺎﻌﺸﻟا ﻦﯿﺗ

3 AG uuuur = AB uuur

َو ' 1

AG = 3AB uuuur uuur

(2 ' 8

GGuuuur= −3ABuuur

(1

ﺔﯿﻋﺎﻌﺸﻟا تﺎﻗﻼﻌﻟا ءﺎﺸﻧﻺﻟ مﺪﺨﺘﺴﻧ :

' 2 , ' 3 , ' 3

B C = AC A A= BA C C = 2BC uuuuur uuuur uuuur uuur uuuur uuur

(3 ﻲﻓ ةاوﺎﺴﻤﻟا ﻦﻣ (2

ﺪﺠﻧ : ' ' 2 ' ' A B = A C uuuuur uuuuur

(2 نأ ﻆﺣﻻ :

( )

2 3 ' 2 3

2 1 ' 2

3 1 ' 3

MA MA MB MB MA MC MC MB MC

 − = −

 − + = − +

 − = −



uuuur uuuur uuuur uuuuur uuuur uuuur uuuuur uuuur uuuur

(2

1 2

2 3 , 3

AG = − AB+ AC BG = −AB+2AC uuuur uuur uuuur uuuur uuur uuuur

(3 نأ ﻆﺣﻻ :

1 2 2

AG = BG uuuur uuur

(1

( )( )

{

^,1 ^{, 4}

}

N = C B − يأ

4 0

NC − NB = uuuur uuur r ﺪﺠﻧ لﺎﺷ ﺔﻗﻼﻋ ماﺪﺨﺳﺎﺑ و

:

3 0

BC + BN = uuur uuuur r

(2

نأ تﺎﺒﺛﻹ ﺔﻨھﺮﺒﻤﻟا ﺲﻔﻧ ﻞﻤﻌﺘﺴﻧ و ﺲﯿﻟﺎﻃ ﺔﻨھﺮﺒﻣ ﻞﻤﻌﺘﺳا عﻼﺿأ يزاﻮﺘﻣ LMJI

ﻹ نأ تﺎﺒﺛ

( )( )( )

{

^,1 ^,1 ^,1

}

O = A B C نأ ﻆﺣﻼﻧ

:

( )( )

{

^{, 3} ^{, 3}

} { ⁽

^{, 2}

^{)( )(}

^,1 ^{, 2}

^{)( )}

^,1

} { ⁽

^{, 2}

⁾⁽

^{, 2}

⁾⁽

^{, 2}

⁾ } { ^{( )( )( )}

^,1 ^,1 ^,1

}

O = L J = A C B C = A B C = A B C

(1

( )( )

{

^{, 2} ^{, 3}

}

L = A − C

(2

( )( )

{

^,1 ^{, 4}

}

K = B C

(1

( )( )

{

^{, 5} ^,6

}

M = B C ،

( )( )

{

^{, 2} ^,5

}

N = A B

( )( )

،

{

^,1 ^{, 3}

}

P = A C

(2

ﻦﻜﺘﻟ ﻊﻃﺎﻘﺗ ﺔﻄﻘﻧ G

(

^NC

) ( )

^BP َو

نأ ﻦھﺮﺒﻧ

( )

G∈ AM

نﻮﻛ ﻦﻋ ﺮﺒﻌﻧ نأ ﻦﻜﻤﯾ ﻦﯿﺘﻄﻘﻨﻟا ﺢﺟﺮﻣ G

َو A M

دﻮﺟو ﻦھﺮﺒﻧ α

ﺚﯿﺣ

( )( )( )

:

{

^, ^{, 5} ^{, 6}

} { ⁽

^,

⁾⁽

^,11

⁾ }

G = A α B C = A α M ﻲﻟﺎﺘﻟﺎﺑ و

( )

G∈ AM

نأ ضﺮﻔﻧ

( )( )( )

{

^, ^{, 5} ^{, 6}

}

G = A α B C ﻲﻤﺴﻧ و

( )( )

{ }

' , , 6

P = A α C

ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ ﺎﻀﯾأ ﻮھ G

( )( )

:

{

^B^{, 5} ^P^',^α ⁺⁶

}

نذإ

( ) ( )

P ∈ AC ∩ BG نذإ

' P =P

نأ ﺎﻤﺑ و

( )( )

{

^,1 ^,3

}

P = A C نﺈﻓ

: α =2

(2 نأ ﻆﺣﻻ 3 :

AI =2AB uuur uuur ﻲﻟﺎﺘﻟﺎﺑو

:

( ) ( )

³₂

CIuur uuur uuur uuur uuur uuur=CA+AI = CB+CD +AI = −BCuuur uuur−AB + ABuuur نأو

: 3 AJ = AD uuur uuuur ﻲﻟﺎﺘﻟﺎﺑ و

( )

³

( )

³ ²

CJuuur uuur uuur=CA+AJ = CBuuur uuur+CD + ADuuuur= −ADuuuur uuuur−DC + ADuuuur= ADuuuur uuuur−DC

(3 نأ ﻆﺣﻻ :

2 CJuuur= − CIuur

(4 (أ ﺎﻨﯾﺪﻟ ﻒﺼﺘﻨﻣ D

[ ]

^AK

( ) (

^AL ^// ^DC

)

َو

نذإ ﻒﺼﺘﻨﻣ C

[ ]

^KL

نأ ﻆﺣﻻ )

[ ]

C ∈ KL ب (

نأ ﻦھﺮﺒﻧ نأ ﻦﻜﻤﯾ( 2

KL= DB uuur uuur عﻼﺿأ يزاﻮﺘﻣ صاﻮﺧ لﺎﻤﻌﺘﺳﺎﺑ

(5)

(1

ﺔﻗﻼﻌﻟا ﻞﻤﻌﺘﺴﻧ 2 :

CDuuur=3CBuuur

(2 ﺔﻗﻼﻌﻟا ﻞﻤﻌﺘﺴﻧ 1 :

AE =3AC uuur uuuur

(3 ةرﻮﺻ ﻮھ F ﮫﻋﺎﻌﺷ يﺬﻟا بﺎﺤﺴﻧﻹﺎﺑ A

BDuuur

(4 نأ ﻦھﺮﺒﻧ ﺔﻘﺑﺎﺴﻟا ﺔﻠﺌﺳﻷا و لﺎﺷ ﺔﻗﻼﻋ لﺎﻤﻌﺘﺳﺎﺑ :

5 DF =3DE uuur uuur

(1 نﻮﻜﺗ نﺎﻛ اذإ ﻂﻘﻓ و اذإ ةدﻮﺟﻮﻣ G

(

^m^²^{+ +}²

) (

^m^²^{+ − ≠}^m ³

)

⁰ : 1 يأ

1 ,2 m∈IR− − 

 

(1 ﺔﻟﺎﺤﻟا P =C ﺪﺠﻧ 4 : β = −3 ( 2 ﺔﻟﺎﺤﻟا 2

PC = AB uuur uuur ﺪﺠﻧ

: 2 β = −15

ﻂﻘﻧ ثﻼﺛ ﺢﺟﺮﻣ

لﺎﻤﻌﺘﺳإ ﻞﻀﻔﯾ تﻻﺎﺤﻟا هﺬھ ﻲﻓ ﺢﺟﺮﻤﻟا ءﺎﺸﻧﻹ ﻊﻤﺠﺘﻟا ﺔﯿﺻﺎﺧ

(1

( )( )( )

{

^,1 ^,1 ^,1

}

G = A B C

(2

( )( )( )

{

^{, 5} ^{, 3} ^{, 1}

}

G = A B − C −

(3

( )( )( )

{

^{, 6} ^{, 6} ^{, 1}

}

G = A B − C −

(4

( )( )( )

{

^{, 5} ^{, 3} ^{, 2}

}

G = A − B C −

(5

( )( )( )

{

^{, 1} ^{, 3} ^{, 2}

}

G = A − B − C (6

( )( )( )

{

^{, 1} ^{, 0} ^{, 4}

}

G = A − B C −

(1 ﻦﯿﺘﻄﻘﻨﻟا ﺮﺒﺘﻌﻧ

( )( )

{ }

1 , 1 , 2

G = A − C ﻒﺼﺘﻨﻣ ﻢﺛ

[

^{G B}¹

]

(2 ﺮﺒﺘﻌﻧ ﻒﺼﺘﻨﻣ I

[ ]

^BC

ﻢﺛ

( )( )

{

^{, 2} ^,1

}

F = I − A

(3

( )( )

{

^{, 2} ^,1

}

G = B C

(4 ﻔﯿﻛ رﺎﯿﺘﺧإ ﻲ

(5 رﺎﯿﺘﺧإ

( )( )

{ }

1 ,3 , 2

I = A B − ﻢﺛ

ﻒﺼﺘﻨﻣ

[ ]

^{I C}¹

(6 J =C

(1

( )( )( )

{

^{, 3} ^{, 4} ^{, 2}

}

G = A − B C

(2

( )( )( )

{

^{, 2} ^{, 5} ^{, 5}

}

G = A B − C −

(3

( )( )( )

{

^{, 2} ^{, 0} ^{, 1}

}

G = A B C −

(4

( )( )( )

{

^{, 0} ^,1 ^,1

}

G = A B C

(5

( )( )( )

{

^,1 ^{, 2} ^{, 4}

}

G = A B − C

ﻰﻟوﻷا ﺔﻟﺎﺤﻠﻟ ﺔﺒﺴﻨﻟﺎﺑ

( )( )( )

:

{

^,1 ^,1 ^{, 1}

}

D = A B C − هﺎﻨﻌﻣ

( )( )( )

{

^,1 ^{, 1} ^{, 1}

}

A= B C − D − هﺎﻨﻌﻣ

( )( )( )

{

^,1 ^{, 1} ^{, 1}

}

B = A C − D − هﺎﻨﻌﻣ

( )( )( )

{

^,1 ^,1 ^{, 1}

}

C = A B D −

(1 ﺊﺸﻨﻧ ﻒﺼﺘﻨﻣ I

[ ]

^AC

G ﻢﺛ ﻒﺼﺘﻨﻣ ﻮھ

[ ]

^IB

(2

(

^{α β γ}^, ^,

) (

^{= −}^{4 , 2 , 1}

)

(1 : نﻷ

( )

1 2+ + − ≠4 0

(2

2 4

AG = − AB+ AC uuuur uuur uuuur ﺠﻟا صاﻮﺧ لﺎﻤﻌﺘﺳﺎﺑ ﺊﺸﻨﻧ و

ﻲﻋﺎﻌﺸﻟا ﻊﻤ

(2

ﺔﻗﻼﻌﻟا لﺎﻤﻌﺘﺳﺎﺑ 3

BK = BC uuur uuur ﺚﯿﺣ )

( )( )

{

^{, 2} ^{, 3}

}

K = B − C و (

ﺔﻈﺣﻼﻤﺑ

نأ ﻒﺼﺘﻨﻣ G

[ ]

^AK

و َو I ﻲﻔﺼﺘﻨﻣ J

[ ]

^AB

[ ]

^AC َو

نأ ﻦﻜﻤﯾ

نأ ﻦھﺮﺒﻧ :

3 IG = IJ uuur uur

(1 ﻂﻘﻨﻟا ءﺎﺸﻧإ ﻦﻜﻤﯾ , ,

K H G : نﻷ

3 2 0

− + ≠ َو

2 1 0

− + ≠

(2 ﻦﻜﻤﯾ لﺎﻤﻌﺘﺳا نأ ﻰﻠﻋ نﺎھﺮﺒﻠﻟ ﺲﯿﻟﺎﻃ ﺔﻨھﺮﺒﻣ :

2 GHuuuur= GKuuur

(6)

(1 ﺮﻄﻗ صاﻮﺧ و ﻲﻋﺎﻌﺸﻟا ﻊﻤﺠﻟا لﺎﻤﻌﺘﺳإ ﻦﻜﻤﯾ لﺎﺷ ﺔﻗﻼﻋ لﺎﻤﻌﺘﺳإ وأ عﻼﺿأ يزاﻮﺘﻣ ي

ﺔﻌﻄﻗ ﻒﺼﺘﻨﻣ صاﻮﺧو

(1 ﻲﻓ ﺔﻨھﺮﺒﻤﻟا ﺔﯿﻋﺎﻌﺸﻟا ﺔﻗﻼﻌﻟا ﻦﻣ (1

ﺞﺘﻨﯾ :

2 0

AB+AC − AI = uuur uuuur uuur r ﮫﻨﻣ و

: ^A⁼

{ (

^I^{, 2}⁻

)( )( )

^B^,1 ^C^,1

}

(1 ءﺎﺸﻧﻹا

(2

( )( )( )

{

^{, 0} ^,1 ^,1

}

I = A B C

( )( )( )

{

^,1 ^{, 2} ^{, 1}

}

J = A B − C −

(1 ﺔﻗﻼﻌﻟا لﺎﻤﻌﺘﺳﺎﺑ ﺊﺸﻨﻧ :

3 IB = BC uur uuur

(2 ﻒﺼﺘﻨﻣ G

[ ]

^AI

نﻷ

( )( )

:

{

^,1 ^,1

}

G = I A

(1 موﺪﻌﻣ ﺮﯿﻏ تﻼﻣﺎﻌﻤﻟا عﻮﻤﺠﻣ نﻷ ﻦﯿﺤﺟﺮﻣ نﻼﺒﻘﺗ ﻦﯿﺘﻠﻤﺠﻟا

(2

ﻦﯿﺗوﺎﺴﻤﻟا ﻊﻤﺠﺑ :

2 0

KB− KA= uuur uuur r 3 0 َو

LC + LA= uuur uuur r ﺠﻧ ﻦﯿﺗوﺎﺴﻤﻟا ﻲﻓ لﺎﺷ ﺔﻗﻼﻋ ماﺪﺨﺘﺳﺎﺑ َو

: ﺪ

4 0

GA GBuuur uuur uuur+ +GC + LGuuur uuuur r−KG = يأ

4 0

GK GL

−uuur+ uuur r= نﻷ )

ﺚﻠﺜﻤﻟا ﻞﻘﺛ ﺰﻛرو G ( ABC

(1 ﺔﯿﺻﺎﺨﻟا لﺎﻤﻌﺘﺳإ ءﺎﺸﻧﻺﻟ ﻦﻜﻤﯾ :

3 3

GMuuuur uuuur uuuur uuuur=AM + BM − CM ﻼﺜﻣ ﺬﺧﺄﻧ َو

: M =A ﺪﺠﻧ

3 3

GAuuur= BAuuur uuur− CA يأ

3 GAuuur= BCuuur

(2 لوﻷا لاﺆﺴﻟا )

3 GAuuur= BCuuur لاﺆﺴﻟا اﺬھ ﻦﻋ ﺐﯿﺠﯾ (

(2

ﻊﯿﻤﺠﺘﻟا ﺔﯿﺻﺎﺧ ﻞﻤﻌﺘﺴﻧ ﺐﺘﻜﻨﻓ

( )( )( )( )

:

{

^,1 ^,1 ^,1 ^{, 3}

}

M = A B C D

يأ

( )( )

:

{

^{, 3} ^,3

}

M = G D نأ يأ

ﻒﺼﺘﻨﻣ G

[ ]

^GD

(3

نأ نﻵا ﺮﺒﺘﻌﻧ

( )( )

:

{

^{, 2} ^{, 4}

}

M = I F يأ

, , M I F ﺔﯿﻣﺎﻘﺘﺳإ ﻲﻓ

(4 نأ نﻵا ﺮﺒﺘﻌﻧ

( )( )

:

{

^{, 4} ^{, 2}

}

M = L P ﺚﯿﺣ

( )( )

:

{

^,1 ^,1

}

P = B C

( )( )

َو

{

^,1 ^{, 3}

}

L= A D ﺔﯿﻘﺒﻟا و

ﺔﺤﺿاو

ﺐﺘﻜﻧ نأ ﻦﻜﻤﯾ ﮫﻧأ ﻆﺣﻻ :

^G^∈

( )

^AI ^نذإ^G ⁼

{ ( )( )

^A^,1 ^I^{, 6}

}

( )( )

ﻢﺛ

{

^{, 2} ^,5

}

G = B J نذإ

( )

G∈ BJ

( )( )

{

^{, 4} ^{, 3}

}

G = C H نذإ

( )

G∈ CH

(1 ﺐﺘﻜﻧ : GBuuur uur uur=GI +IB َو

GCuuur uur uur=GI +IC

(2 ﻓ ﻖﺑﺎﺴﻟا لاﺆﺴﻟا ﻲﻓ ةدﻮﺟﻮﻤﻟا ةاوﺎﺴﻤﻟا ضﻮﻌﻧ ﻲ

ﺔﻗﻼﻌﻟا :

2GA GBuuur uuur uuur r+ +GC =0 ﺪﺠﻧ

: 0 GA GIuuur uur r+ = نأ يأ

( )( )

{

^,1 ^,1

}

G = A I

(3 ضﻮﻌﻧ

ﻦﯿﺘﻄﻘﻨﻟا ﻦﯿﺘﻄﻘﻨﻠﻟ ﻦﯿﻠﻣﺎﻌﻤﻟا عﻮﻤﺠﻤﺑ ﻞﻘﺜﻤﻟا ﺎﻤﮭﻔﺼﺘﻨﻤﺑ ﻞﻣﺎﻌﻤﻟا ﺲﻔﻨﺑ ﻦﯿﺘﻠﻘﺜﻤﻟا

(1 ﻤﯾ ﺐﺘﻜﻧ نأ ﻦﻜ

( )( )

:

{

^{', 2} ^{, 3}

}

H = C J نأ يأ

:

( )( )( )( )

{

^,1 ^,1 ^,1 ^{, 2}

}

H = A B B C نﻷ

( )( )

{ }

( )( )

{ }

' ,1 ,1

,1 , 2

C A B

J B C

 =

 = 

ﻲﻟﺎﺘﻟﺎﺑ و

( )( )( )

:

{

^,1 ^{, 2} ^{, 2}

}

H = A B C (2

ﺲﯿﻟﺎﻃ ﺔﻨھﺮﺒﻣ ﻞﻤﻌﺘﺳإ نأ ﻆﺣﻻ )

( ) (

^IJ ^// ^{B C}^' ^'

)

(

(2

نأ ﻆﺣﻻ

( )( )

:

{

^{, 3} ^{, 2}

}

G = I C − ﻦﻜﻟ

( )( )

:

{

^{, 2} ^,1

}

I = A B

ﻲﻟﺎﺘﻟﺎﺑَو

( )( )( )

:

{

^{, 2} ^,1 ^{, 2}

}

G = A B C −

(7)

(3 أ

( )( )

(

{

^,1 ^{, 2}

}

L= B c − ﻲﻟﺎﺘﻟﺎﺑ و

( )

L∈ BC ﻧ نأ ﻦﻜﻤﯾ و

( )( )

ﺐﺘﻜ

{

^{, 2} ^{, 1}

}

G = A L − ﻲﻟﺎﺘﻟﺎﺑ و

( )

L∈ AG

ﺺﻠﺨﺘﺳا

ب نأ ﺎﻤﺑ ( L =H َو 2 BL= BC uuur uuur نﺈﻓ

: 2 k =

(1 نﺎﻛ اذإ ﮫﻧأ ﻢﻠﻋا ﺚﻠﺜﻣ ﻞﻘﺛ ﺰﻛﺮﻣ G

نﺈﻓ : 2 AG = GI uuuur uur ﺎﻨﯾﺪﻟ و

2 GHuuuur= GIuur

(2

( ) ( )

²

HB+HC = HI +IB + HI +IC = HI =HG uuur uuuur uuur uur uuur uur uuur uuuur

(3

( )

2

HA= HB +HC uuur uuur uuuur ﻲﻟﺎﺘﻟﺎﺑ و

( )( )( )

:

{

^,1 ^{, 2} ^{, 2}

}

H = A B − C −

(1 ةاوﺎﺴﻤﻟا ﻦﻣ 1 :

AI =3AC uuur uuuur ﺪﺠﻧ

: 2IAuur uur+IC =0

(2 ةاوﺎﺴﻤﻟا ﻦﻣ 1

BG =3BI uuur uuur ﺪﺠﻧ

:

2GBuuur uur r+GI =0

(3

(أ ةاوﺎﺴﻤﻟا ﻊﻣ لﺎﺷ ﺔﻗﻼﻋ مﺪﺨﺘﺳا 2IAuur uur+IC =0

(ب جﺮﺨﺘﺳا ةاوﺎﺴﻤﻟا ﻦﻣ كﺮﺘﺸﻣ ﻞﻣﺎﻋ 3

(3 أ- (

(2 ﺐﺘﻜﻧ

( )( )( )

:

{

^{, 2} ^,1 ^{, 3}

} { ^{( )(}

^{, 3} ^{, 3}

⁾ }

H = I C D = G D يأ

( )

H ∈ DG

(3

( )( )

{ } { ^{( )( )} }

{

^,1 ^{, 3} ^,1 ^,1

} { ⁽

^{, 4}

^{)( )}

^{, 2}

}

H = A D B C = K J

( )

يأ H ∈ JK

(4

( )( )

{ } { ^{( )(} ⁾ }

{

^,1 ^,1 ^,1 ^{, 3}

} { ^{( )( )}

^{, 2} ^{, 4}

}

H = A B C D = I L

( )

يأ H ∈ IL

(5

ﺺﻠﺨﺘﺳا

(1

ﻞﺟأ ﻦﻣ ﻞﻜﺸﻟا 1

k =3

(2 تﺎﻗﻼﻌﻟا ﻦﻣ

1 1 1 :

, ,

3 3 3

BI = BC CJ = CA AL= AB uuur uuur uuur uuur uuur uuur

ﺪﺠﻧ 3GIuur=2GBuuur uuur uuur+GC , 3GJ =2GCuuur uuur uuur+GA, 3GL =2GA GBuuur uuur+ و

ﺪﺠﻧ فﺮﻃ ﻰﻟإ ﺎﻓﺮﻃ ﺔﺛﻼﺜﻟا تاوﺎﺴﻤﻟا ﻊﻤﺠﺑ :

0 GIuur uuur uuur r+GJ +GL =

(3 ﺮﻣ ﻲھ G ﺚﻠﺜﻤﻟا ﻞﻘﺛ ﺰﻛ IJL

(1 1 ﻦﻣ

AK =3AB uuuur uuur ﺪﺠﻧ

: 2KAuuur uuur r+KB =0 : يأ

( )( )

{

^{, 2} ^,1

}

K = A B

ﻦﻣ و 2 :

BI = 3BC uuur uuur ﺪﺠﻧ

: 2IBuur uur r+IC =0 يأ

( )( )

{

^{, 2} ^,1

}

I = C B

(2 و (3 ﺲﯿﻟﺎﻃ ﺔﻨھﺮﺒﻣ لﺎﻤﻌﺘﺳا ﻦﻜﻤﯾ

ﻲﻋﺎﺑﺮﻟا -

نﺎﻔﺻﺎﻨﺘﻣ هاﺮﻄﻗ نﻷ عﻼﺿأ يزاﻮﺘﻣ EKJI

(1

( )( )

{

^,1 ^{, 2}

}

I = A B

( )( )

،

{

^{, 2} ^{, 3}

}

J = B C

(2 ﺐﺘﻜﻧ نأ ﻦﻜﻤﯾ :

( )( )( )

{

^,1 ^{, 2} ^{, 3}

} { ^{( )( )}

^,3 ^,3

}

H = A B C = I C

يأ ﻒﺼﺘﻨﻣ H

[ ]

^IC

نأ ﺎﻤﺑ َو ﻒﺼﺘﻨﻣ G

[ ]

^IC

نﺈﻓ : G =H (3 نأ ﻆﺧﻻ :

( )( )

{

^,1 ^{, 5}

}

G =H = A J

(1 نأ ﺎﻤﺑ

( )( )

:

{ } ⁽ ⁾

{

^{, 2} ^{, 1} ^{, 2}

} { ⁽

^{, 3}

⁾⁽

^{, 2}

⁾ }

G = A − B − C = I − C

ﻂﻘﻨﻟا نﺈﻓ , ,

G J A ﺘﺳا ﻲﻓ

ﺔﯿﻣﺎﻘ

(8)

ﻂﻘﻨﻠﻟ ﺔﺒﺴﻨﻟﺎﺑ - , ,

G I C نأ ﻆﺣﻻ :

( ) ( { )( ) }

{

^{, 2} ^{, 1} ^{, 2}

} { ⁽

^{, 2}

^{)( )}

^,1

}

G = A − B − C = A − J

(2 ﻦﻣ (1

( )

G∈ AJ

( )

َو G∈ CI

( 3 نأ ﻆﺣﻻ :

ﻒﺼﺘﻨﻣ A

[ ]

^GJ

نأ َو ﻒﺼﺘﻨﻣ C

[ ]

^BJ

ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ

(1

( ) ( )

²

MA+MB = MI +IA + MI +IB = MI uuuur uuuur uuur uur uuur uur uuur

(2 MA+MB =AB uuuur uuuur

هﺎﻨﻌﻣ

2 MI =AB uuur

َو ﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ E

ﺎھﺰﻛﺮ ﺎھﺮﻄﻗ ﻒﺼﻧ َو I

2 R = AB

(1 ءﺎﺸﻧﻹا

(2

2 4

MA+ MB = −MA+ MB uuuur uuuuur uuuuur uuuuur يأ

MG =MH

(3

( )

^E

ﺔﻌﻄﻘﻟا رﻮﺤﻣ ﻲھ

[ ]

^GH

(1 ﺔﻗﻼﻌﻟا ﻦﻣ 3

AK = −2AB uuuur uuur ﺪﺠﻧ

: 5KAuuur uuur r−3KB =0

( )( )

يأ

{

^{, 5} ^{, 3}

}

K = A B −

(2 5MAuuuur uuuuur−3MB =AB ﺊﻓﺎﻜﺗ

2MKuuuur =AB 2 يأ

MK = AB

2 و ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ E

ﺎھﺰﻛﺮﻣ ﺎھﺮﻄﻗ ﻒﺼﻧ َو K

2 R =AB

(2 أ نﺎﻋﺎﻌﺸﻟا ( 2MAuuuur uuuur+MB

َو ABuuur ﺎﯿﻄﺧ نﺎﻄﺒﺗﺮﻣ

هﺎﻨﻌﻣ //

MG AB uuuur uuur

1 َو ﻤﻟا ﻮھ E ﻢﯿﻘﺘﺴ

( )

^AB

ب ( 2MAuuuur uuuuur+MB =AB هﺎﻨﻌﻣ

3 MG =AB َو

E2

ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ ﺎھﺮﻄﻗ ﻒﺼﻧ َو G

3 R = AB

ـﺟ ( 2MAuuuur uuuuur+MB =3MA هﺎﻨﻌﻣ

MG =MA

3 َو ﺔﻌﻄﻘﻟا رﻮﺤﻣ ﻲھ E

[ ]

^GA

(1 َو (2 مﺪﻘﺗ ﺎﻤﻛ ﻞﻜﺸﻟا ﺄﺸﻨﯾ

:(3 3MA MBuuuur uuuuur− = −2MCuuuur+4MDuuuuur

ﺊﻓﺎﻜﺗ MG =MK ﻄﻘﻟا رﻮﺤﻣ ﻲھ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣو

[ ]

^GA ﺔﻌ

(1 مﺪﻘﺗ ﺎﻤﻛ ﻞﻜﺸﻟا ﺄﺸﻨﯾ

(2 ﺔﻋﻮﻤﺠﻤﻟا ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ E

ﺎھﺮﻄﻗ ﻒﺼﻧ َو C 2

(3 ﺔﻋﻮﻤﺠﻤﻟا '

ﺔﻌﻄﻘﻟا رﻮﺤﻣ ﻲھ E

[ ]

^CD

ﺘﻟا ﻦﯾﺮﻤ : 70 (1 ﺔﻟاﺪﻟا ﺔﻄﻘﻧ ﻞﻜﺑ ﻖﻓﺮﺗ f ﺔﻄﻘﻨﻟا يﻮﺘﺴﻤﻟا ﻦﻣ M

' ﺚﯿﺣ M : ' 2 MM = MG uuuuuur uuuur نأ يأ

' 0 GMuuuur uuuuur r+GM = ﺔﻟاﺪﻟا نأ يأ

ﻄﻘﻧ ﻞﻜﺑ ﻖﻓﺮﺗ f ﺔ

ﺎﮭﺗﺮﯿﻈﻧ يﻮﺘﺴﻤﻟا ﻦﻣ M '

ﻰﻟإ ﺔﺒﺴﻨﻟﺎﺑ M G

(2 (أ ' ur uuuuuur=MM هﺎﻨﻌﻣ

'

MM =BA CA+ uuuuuur uuur uuur ﺔﻟاﺪﻟا و

ﺔﻄﻘﻨﻟﺎﺑ ﻖﻓﺮﺗ g ﺔﻄﻘﻨﻟا M

' ﮫﻋﺎﻌﺷ بﺎﺤﺴﻧﻹﺎﺑ ﺎﮭﺗرﻮﺻ M ur

ب ﺔﻋﻮﻤﺠﻤﻟا ( ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ E

عﺎﻌﺸﻟا ﺔﻠﯾﻮﻃ ﺎھﺮﻄﻗ ﻒﺼﺗ و G 2

uuur

(9)

:71 (1

2 2

MA MB− − MC = − MG uuuur uuuur uuuur uuuur

(2

( )

M ∈ E ﺊﻓﺎﻜﺗ

2MG 4

− uuuuur = ﺊﻓﺎﻜﺗ

2 MG = uuuur

(3 ﺔﻋﻮﻤﺠﻤﻟا

( )

^E

ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ ﺎھﺮﻄﻗ ﻒﺼﺗ و G

2

: 72

(1 ﻲﻋﺎﺑﺮﻟا ناﺮﻄﻗ ﮫﯿﻓ ABCG

نﺎﺴﯾﺎﻘﺘﻣ نﺎﻌﺑﺎﺘﺘﻣ نﺎﻌﻠﺿ َو نﺎﻔﺻﺎﻨﺘﻣ

(2 أ ( ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ E G

ﺎھﺮﻄﻗ ﻒﺼﺗ و 5 3

R = 2 ب ﻒﺼﺘﻨﻣ (

[ ]

^AC

ﺔﻄﻘﻧ ﻦﻣ نﻷ E 5 3 GI = 2

: 73

(1 ﺐﺘﻜﻧ

( )( )

:

{ } ⁽ ⁾

{

^{, 1} ^{, 1} ^{, 4}

} { ⁽

^{, 4}

⁾⁽

^{, 2}

⁾ } { ⁽

^{, 2}

⁾⁽

^{, 1}

⁾ }

G = B − C − A = A I − = A I −

(2

( )( )( )

{

^{, 2} ^,1 ^,1

}

A= G B C

(3 ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ ﺎھﺮﻄﻗ ﻒﺼﻧ و A

2 R =BC

: 74 (1 ﺸﻨﻧ مﺪﻘﺗ ﺎﻤﻛ ﺊ

(2

ﺔﮭﺟ ﻦﻣ ﺐﺘﻜﻧ

( ) ( { )( ) }

:

{

^,1 ^{, 4} ^{, 2}

} { ^{( )( )}

^,1 ^{, 2}

}

G = A B C − = A J نأ يأ

( )

G∈ AJ

ىﺮﺧأ ﺔﮭﺟ ﻦﻣ َو

( )( )

:

{ } ⁽ ⁾

{

^,1 ^{, 4} ^{, 2}

} { ^{( )(}

^,5 ^{, 2}

⁾ }

G = A B C − = I C − نأ يأ

:

( )

G∈ CI

(3

ﺪﺠﻧ : MI =MJ ﺔﻌﻄﻘﻟا رﻮﺤﻣ ﻲھ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ و

[ ]

^IJ

(4

ﺪﺠﻧ 4 :

MG =3AC ﺎھﺰﻛﺮﻣ ةﺮﺋاﺪﻟا ﻲھ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ و

ﺎھﺮﻄﻗ ﻒﺼﻧ و G 4

R =3AC

: 75 (1 ﻲﻤﺴﻧ

( )( )( )

{ }

1 , 3 ,5 , 2

G = A B C َو

( )( )

{ }

2 , 3 , 2

G = A C ﺪﺠﻧ

1 2 :

MG =MG ﺔﻋﻮﻤﺠﻤﻟاو

E1

ﻲھ

ﺔﻌﻄﻘﻟا رﻮﺤﻣ

[

^{G G}¹ ²

]

(2

: ﺪﺠﻧ

1

3 MG =10AC uuuuur uuuur ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ و

E2

ﺔﻄﻘﻨﻟا ﻲھ ةاوﺎﺴﻤﻟا هﺬھ ﻖﻘﺤﺗ ﻲﺘﻟا M

(3

ﺪﺠﻧ

2 : 5AB 2AC 10MG

− uuur+ uuuur = ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ و

E3

ﺎھﺰﻛﺮﻣ ةﺮﺋاﺪﻟا ﻲھ G2

ﺎھﺮﻄﻗ ﻒﺼﻧ و

5 2

10 AB AC

R − +

=

uuur uuuur

:76 (1 ﺐﺘﻜﻧ : 0 a DA DBuuur uuur− +a DCuuuur r= يأ

( )

:

a DAuuur uuuur uuur+DC =DB ﻦﻜﻟ

نذإ ﻞﯿﻄﺘﺴﻣ ABCD 1

a =

(2

نأ ﻆﺣﻼﻧ

( )

:

u M =MA MBuuuur uuuur uuuur uuuur− +MC =MD َو

( )

v M =CDuuur uuuur+AD

نﺎﻋﺎﻌﺸﻠﻟ ( )

َو u M

( )

هﺎﻨﻌﻣ ﺔﻠﯾﻮﻄﻟا ﺲﻔﻧ v M MD = CDuuur uuuur+AD

ةﺮﺋاﺪﻟا ﻲھ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ و

ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ﺔﻄﻘﻨﻟا ﻞﻤﺸﺗ و D

B

77 (1

( )( )( )

{ }

' , 1 ,1 , 2

M = A − B M ﺊﻓﺎﻜﺗ

' 1

MM = 2AB =AI uuuuuur uuur uuur ﺺﻠﺨﺘﺳا

(2

( )( )( )

{ }

'' ,1 ,1 , 1

M = A B M − ﺊﻓﺎﻜﺗ

'' 0 IM +IM = uuur uuuur r ﺺﻠﺨﺘﺳا

(3 ﺔﻄﻘﻨﻟا ﺢﺴﻤﺗ ﺎﻣﺪﻨﻋ ﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا M

ﺎھﺰﻛ ﻞﻤﺸﺗ و A

I

أ ﺔﻄﻘﻨﻟا (

' ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﺢﺴﻤﺗ M ﻞﻤﺸﺗ و I

A

(10)

ب ﺔﻄﻘﻨﻟا (

'' ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﺢﺴﻤﺗ M ﻞﻤﺸﺗ و B

I

: 78 (2 (أ 2MAuuuur uuuur uuuur+MB+MC =4MGuuuur

ب ﺐﺘﻛأ (

: MB =MA+AB uuuur uuuur uuur َو

MC =MA+AC uuuur uuuur uuuur

ـﺟ ﺸﻟا ﺮﻈﻧأ (

د ﻞﻜ (

4 2 , 2

AD = AG =

(3

أ ﺪﺠﻧ ( : 2 MG = ﺎھﺰﻛﺮﻣ ةﺮﺋاد ﺎھﺮﻄﻗ ﻒﺼﻧ و G

2 R =

: 79 (1 نأ ﺎﻤﺑ

(

¹

) (

¹

) (

³ ¹

)

¹

k + k + + k − + − k + = نﺈﻓ

ـﻟ ﺔﻤﯿﻗ ﻞﻛ ﻞﺟأ ﻦﻣ ﺔﻓﺮﻌﻣ G k

(2

نأ ﻆﺣﻻ ﻲﻟﺎﺘﻟﺎﺑ و عﻼﺿأ يزاﻮﺘﻣ ABCD

: AC =AB+AD uuuur uuur uuuur نذإ

: ^A⁼

{ ( )(

^B^,1 ^C^{, 1}⁻

)( )

^D^,1

}

لﺎﺷ ﺔﻗﻼﻋ ماﺪﺨﺳﺎﺑ -

ﺔﻄﻘﻨﻟا ﻞﻤﻌﺘﺳأ ) ﺔﻌﺷﻷا ﻲﻓ A

, ,

GB GC GDuuur uuur uuur ﺔﻗﻼﻌﻟا ماﺪﺨﺘﺴﻏ و (

:

(

¹

) (

¹

) (

³ ¹

)

⁰

kGAuuur+ k + GBuuur+ k − GCuuur+ − k + GDuuur r= ﺪﺠﻧ

:

(

³

) ( )

GAuuur+k ABuuur uuuur uuuur+AC − AD + ABuuur uuuur uuuur−AC +AD ﺪﺠﻧ ﻲﻜﻟ

: 2

AG = k DB uuuur uuur

(3

ﮫﮭﯿﺟﻮﺗ عﺎﻌﺷ يﺬﻟا ﻢﯿﻘﺘﺴﻤﻟا ﻲھ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ DBuuur

ﺔﻄﻘﻨﻟا ﻞﻤﺸﯾ و A

: 80 (1 ﻦﯿﺘﻄﻘﻨﻟا مﺪﻘﺗ ﺎﻤﻛ ﺊﺸﻨﻧ

( )( )( )

:

{ }

1 , 2 ,1 , 1

G = A B C −

( )( )( )

َو

{ }

1 , 2 , 1 ,1

G₋ = A B − C ﺔﻄﻘﻨﻟا َو

I

(2 ﺔﻄﻘﻨﻟا Gk

ك نﻷ

(

^k^²^{+ +}¹

) ( ) ( )

^k ^{+ −}^k ^≠⁰

ﻲﻘﯿﻘﺣ دﺪﻋ ﻞﻛ ﻞﺟأ ﻦﻣ k

ةاوﺎﺴﻤﻟا ﻲﻓ لﺎﺷ ﺔﻗﻼﻋ ماﺪﺨﺘﺳﺎﺑ -

(

k^²+¹

)

G Auuuurk =k G Cuuuurk −k G Buuuuurk : ﺔﻄﻘﻨﻟا )

ﻲﻓ B G Cuuuurk

ﺪﺠﻧ (

² 1

k

AG k BC k

= − +

uuuuur uuur

(3 ﺖﻘﺒﻄﻧا اذإ ﻰﻠﻋN

Gk

نﺈﻓ Gk

ﻰﻠﻋ ﻊﻘﯾ

( )

^BC

ﺔﻄﻘﻨﻟا ﻞﻣﺎﻌﻣ ﺬﺋﺪﻨﻋ نﻮﻜﯾ و نأ يأ موﺪﻌﻣ A

:

² 1 0 k + = ﺔﻋﻮﻤﺠﻤﻟا ﻰﻠﻋ ﻞﯿﺤﺘﺴﻣ اﺬھ َو

IR

(4 ﺔﻟاﺪﻟا نأ ﻆﺣﻻ لﺎﺠﻤﻟا ﻰﻠﻋ ﺔﺼﻗﺎﻨﺘﻣ f

[

^{− +}^{1, 1}

]

ﻲھ ىﺮﺒﻜﻟا ﺔﯾﺪﺤﻟا ﺔﯾﺎﮭﻨﻟا ، 1,1

2

− 

 

 

ﺔﯾﺪﺤﻟا ﺔﯾﺎﮭﻨﻟا و

ﻲھ ىﺮﻐﺼﻟا 1, 1

2

 − 

 

 

(5 ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ Gk

ﺔﻤﯿﻘﺘﺴﻤﻟا ﺔﻌﻄﻘﻟا ﻲھ

[

^{G G}−¹ ¹

]

يزاﻮﯾ يﺬﻟا ﻢﯿﻘﺘﺴﻤﻟا ﻦﻣ BCuuur

ﻞﻤﺸﯾ َو A

: 81 (1 ﺚﻠﺜﻤﻟا ﻲﻓ ﻢﺋﺎﻗ ABC

B ( 2

( )

^Γ¹

ـﻟ يزاﻮﻤﻟا ﻢﯿﻘﺘﺴﻤﻟا ﻮھ ACuuuur

ﻞﻤﺸﯾ و

( )( )( )

{

^,1 ^{, 2} ^,1

}

G = A B C

(3

( )

^Γ²

ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ

( )( )( )

{

^,1 ^{, 2} ^,1

}

G = A B C ﻧ و

ﺎھﺮﻄﻗ ﻒﺼ 1

R =4AC

( 4

أ نأ ﻆﺣﻻ ( :

2

MA− MB+MC =BA+BC uuuur uuuur uuuur uuur uuur َو

CAuuur = BAuuur uuur+BC

ب ﺔﻄﻘﻨﻟا ضﻮﻌﻧ (

ﺔﻄﻘﻨﻟﺎﺑ M ةاوﺎﺴﻤﻟا ﻲﻓ B

(4 ـﻟ ﺔﺒﺴﻨﻟﺎﺑ '

ﺾﯾﻮﻌﺘﻟا ﺪﻌﺑ و ﮫﻧأ ﻆﺣﻻ B

' ' 0

B A+B C = uuuuur uuuuur r َو

2BBuuuur' = ACuuuur

ـﺟ (

: 82 (1 ﺐﺘﻜﻧ

( )( )

:

{ } ^{( )}

{

^,1 ^{, 2} ^{, 3}

} { ^{( )( )}

^{, 3} ^{, 3}

}

G = A B C = J C نأ يأ

( )

G∈ JC ﻢﺛ

ﻒﺼﺘﻨﻣ G

[ ]

^JC

(2 نﺎﻋﺎﻌﺸﻟا

2 3

MA+ MB+ MC uuuur uuuur uuuur MA+MC َو

uuuur uuuur هﺎﻨﻌﻣ ﺎﯿﻄﺧ نﺎﻄﺒﺗﺮﻣ

//

MG MI uuuur uuur : ﺚﯿﺣ

ﻒﺼﺘﻨﻣ

[ ]

^AC

ﺔﻋﻮﻤﺠﻤﻟا و

( )

^E

ﻢﯿﻘﺘﺴﻤﻟا ﻲھ

( )

^IG

(3

ﺔﻄﻘﻨﻟا لﺎﻤﻌﺘﺳا ﻚﻟﺬﻟ ﻦﻜﻤﯾ ﻒﺼﺘﻨﻣ K

[ ]

^AB

نأ ﺔﻈﺣﻼﻤﺑ و ﻒﺼﺘﻨﻣ G

[ ]

^JC

(11)

: 83 (1 نأ ﻆﺣﻻ

( )( )

:

{ } ⁽ ⁾

{

^,1 ^,1 ^{, 2}

} { ^{( )(}

^{, 2} ^{, 2}

⁾ }

G = A C B = I B

ﺚﯿﺣ :

ﻒﺼﺘﻨﻣ I

[ ]

^AC

ﻲﻟﺎﺘﻟﺎﺑ و ﻒﺼﺘﻨﻣ G

[ ]

^IB

(2

( )

^E¹

ﺎھﺰﻛﺮﻣ ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ ﺎھﺮﻄﻗ ﻒﺼﻧ و G

4 R =AC

(3 أ نأ ﻆﺣﻻ ( :

2

NA− NB+NC =BA+BC uuur uuur uuuur uuur uuur ﺔﻄﻘﻨﻟا ضﻮﻌﻧ ﻖﻘﺤﺘﻠﻟ

ﺔﻄﻘﻨﻟﺎﺑ N B

ب ﺔﻋﻮﻤﺠﻤﻟا (

( )

^E²

ﺎھﺰﻛﺮﻣ ةﺮﺋاﺪﻟا ﻲھ ﺔﻄﻘﻨﻟا ﻞﻤﺸﺗ و G

B

: 84 (1 ﺔﻋﻮﻤﺠﻤﻟا

( )

^∆

ﺔﻌﻄﻘﻟا رﻮﺤﻣ ﻲھ

[

^{G G}¹ ²

]

ﺚﯿﺣ

( )( )( )

:

{ }

1 ,1 ,1 ,1

G = A B C َو

( )( )

{ }

2 ; 4 , 1

G = D E −

(2 نأ ﻆﺣﻻ :

MD −ME =ED uuuur uuuur uuur رﺎﺒﺘﻋﺎﺑ و

( )( )( )

{ }

3 ,1 , 1 ,1

G = A B − C ﺔﻋﻮﻤﺠﻤﻟا

( )

^Γ

ﻲﺘﻟا ةﺮﺋاﺪﻟا ﻲھ

ﺎھﺰﻛﺮﻣ ﺔﻄﻘﻨﻟا G3

ﺎھﺮﻄﻗ ﻒﺼﻧ و ED

: 85 (1

( )( )( )

{

^,3 ^{, 2} ^{, 4}

}

J = A B − C

(2 نأ ﻦھﺮﺒﻧ 4 :

AC =3GC uuuur uuur

(3 ﺎھﺰﻛﺮﻣ ةﺮﺋاﺪﻟا ﻲھ ﺎھﺮﻄﻗ ﻒﺼﻧ و I

IA

: 86 (1 نﻮﻜﯾ Gk

نﺎﻛ اذإ ﻂﻘﻓو اذإ

{ }

⁰ : k∈IR−

(2 ةاوﺎﺴﻤﻟا و لﺎﺷ ﺔﻗﻼﻋ ﻞﻤﻌﺘﺴﻧ

:

k k k 0

kG A G B G Cuuuur uuuuur uuuur r− + =

(2 ﻲﻋﺎﺑﺮﻟا ABCG1

نﻷ عﻼﺿأ يزاﻮﺘﻣ

1 : AG =BC uuuur uuur

(3 هﺪﺣ ﻢﯿﻘﺘﺴﻣ ﻒﺼﻧ ﻮھ G1

يزاﻮﯾ و

( )

^BC

: 87 (1 Gm

نﻷ دﻮﺟﻮﻣ

( ) (

²^m ^{+ −}¹ ^m

) (

^{+ −}² ^m

)

^≠⁰ : ﻲﻘﯿﻘﺣ دﺪﻋ ﻞﻛ ﻞﺟأ ﻦﻣ

m

(2

( )( )( )

{ }

1 , 2 , 0 ,1

G = A B C ﻦﻣ ﺚﻠﺜﻟا ﻲﻓ

[ ]

^AC

ﻦﻣ ﺎﺒﯾﺮﻗ A

(3

1 2

3 3

m

m m

AG = − AB + − AC

uuuuur uuur uuuur

(4 ﺔﻨھﺮﺒﻤﻟا ﺔﻗﻼﻌﻟا ﻞﻤﻌﺘﺴﻧ ﻲﻓ

(3 لﺎﺷ ﺔﻗﻼﻋ و

(4 ـﻟ يزاﻮﻤﻟا ﻢﯿﻘﺴﻤﻟا ﻲھ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ADuuuur

ﻞﻤﺸﯾ و G1

ﺢﺟﺮﻤﻟا تﺎﯿﺛاﺪﺣإ

88 : (1

( )( )( )

{

^{, 2} ^{, 3} ^{, 5}

}

G = A B − C − ﻲﯿﺛاﺪﺣإ نﻮﻜﺗ و

ﻢﻠﻌﻤﻟا ﻲﻓ G

(

^{A AB AC}^,^{uuur uuuur}^,

)

1 5 : 2 6, G 

 

 

ﺔﻈﺣﻼﻣ ﺔﻘﯾﺮﻄﻟا ﺲﻔﻨﺑ لوﺎﻨﺘﻧ :

(2 َو (3

: 89 (2

( )

^1,0

G ( 3 ب ـﻟ ﻦﯿﺘﯿﻤﻠﺴﻟا ﻦﯿﺘﺒﻛﺮﻤﻟا بﺎﺴﺤﺑ ( ABuuur

َو ACuuuur (4

( )( )

{

^{, 6} ^,1

}

C = A − B

: 90 (2 5, 0 M 2 

 

 

َو

(

^{0 , 5}

)

N ( 3 أ 3 ( 2, 2 I  

 

 

ب 2 (

MI =5MN uuur uuuur ـﺟ

( )( )

(

{

^,3 ^{, 2}

}

I = M N

: 91 (2 2, 1 G−3 

(3

(

^{5 , 4}

)

H − (4 ﺎﻘﺘﺳا ﻲﻓ ﺖﺴﯿﻟ ﺔﯿﻣ

: 92 (2

(

^{5, 6}

)

G − (3

3 14

5, 5 K − 

(4

( )( )( )

{

^{, 3} ^{, 2} ^{, 4}

} { ⁽

^,5

⁾⁽

^{, 4}

⁾ }

G = A − B − C = K C نأ يأ

( )

G∈ KC

ﺪﺠﻧ بﺎﺴﺤﻟﺎﺑ

( )

¹¹⁷ :

CG uuur

⁻

َو

7 5 11

5

CK  

⁻

    uuur

ﮫﻨﻣ و 1 :

CGuuur=5CKuuur

(12)

: 93 (1 : نﻷ 3 7+ ≠0

(2

3 7

10 10

OGuuur= OAuuur+ OBuuur

( 3

10 4 4

G

−

   

 

: 94 ( 2

(

^{5 ,} ⁶

)

G −

( 3 ﻢﯿﻘﺘﺴﻤﻟا ﺮﻤﯾ

( )

^BG

ﻢﻠﻌﻤﻟا أﺪﺒﻤﺑ اذإ O

نﺎﻛ اذإ ﻂﻘﻓو :

//

OB OGuuur uuur

: 95 (2

(

^{2 , 6}

)

H (3 2 ,13 G 3 

 

 

(4 ضﺮﻔﺑ : 1 0 x + ≠ ﺔﻠﻤﺠﻟا ﻞﺤﻧ و

:

2 1

1

1 5 5

1 2

x x x x

 − =

 + +

 =

 +

 َو دﻮﺟﻮﻣ ﺮﯿﻏ x

: 96 (1 هﺎﻨﻌﻣ عﻼﺿأ يزاﻮﺘﻣ BCDE

BC =ED uuur uuur

(

^{4 , 1}

)

يأ E −

(2

5 1

3, 3 G − 

(3 ﺎﻨﯾﺪﻟ

( )( )( )( )

:

{

^,1 ^,1 ^,1 ^,1

}

L= B C D E ﮫﻨﻣ و

(

^{1 ,} ²

)

: L − نأ ﻦھﺮﺒﻧ و

: 3 LA= LG uuur uuur

(4 أ َو ﺔﯿﻠﯿﻠﺤﺘﻟا ﺔﺳﺪﻨﮭﻟا ﻲﻓ ﻲﻋﺎﻌﺸﻟا ﻊﻤﺠﻟا صاﻮﺧ ﻞﻤﻌﺘﺳا ( ﺚﻠﺜﻤﻟا ﻞﻘﺛ ﺰﻛﺮﻣ ﻮھ G

ABD

(5 ﺐﺘﻜﻧ

( ) ( )( ) { } { ^{( )( )} }

:

{

^{, 2} ^,1 ^,1 ^,1 ^,1

} { ⁽

^{, 2}

^{)( )( )}

^{, 2} ^{, 2}

} { ^{( )( )( )}

^,1 ^,1 ^,1

}

G = A B C D E = A I J = A I J

: 97 (1

( )

' 4, 2 َو B

( )

^2,1

K (2

(

^,

)

^{1 3}^, α β 4 4

=   (3

2 4, J 3 3

 

 

(4 نأ ﻆﺣﻻ 3

AC = − IJ uuuur uur

ﺔﻟﺎﻄﻌﻟا ﺰﻛﺮﻣ

: 98 ﺐﺘﻜﻧ : 100GAuuur+xGBuuur r=0 نأ ﻆﺣﻼﻧ و

5 : GBuuur= 2AGuuuur ﺪﺠﻧ

:

B 40 M = g

: 99

( )( )( )

{

^,1 ^',1 ^,16

} { ^{( )(}

^{, 2} ^,16

⁾ } { ^{( )( )}

^,1 ^,8

}

G = H H O = I O = I O

ﺚﯿﺣ : ﻒﺼﺘﻨﻣ I

[

^HH ^'

]

ﮫﺌﺸﻨﻧ و

ةاوﺎﺴﻤﻟا ماﺪﺨﺘﺳﺎﺑ 8 :

IG =9IO uuur uur ﺔﻓﺎﺴﻤﻟا بﺎﺴﺤﻟ و

نأ ﻆﺣﻼﻧ OG

( )

sin 52, 5 OI =OH ⋅ ° َو

1 OG =9OI

: 100 ﻢﻠﻌﻤﻟا ﻲﻓ ﺮﺒﺘﻌﻧ

(

^{A i}^{, ,}^{r ur}^j

)

ﻂﻘﻨﻟا

( ) (

^{0, 0 ,} ^{0 ,18 ,}

) (

^{13 , 18 ,}

) (

^{25 , 0}

)

:

A B C D

ﺰﻛﺮﻣ ﻲﯿﺛاﺪﺣإ ﺐﺴﺤﻧ و

ﻂﻘﻨﻟا هﺬﮭﻟ ﺔﯾوﺎﺴﺘﻤﻟا تﺎﻓﺎﺴﻤﻟا

: 101 ﻰﻟوﻷا ﺔﻘﯾﺮﻄﻟا :

(1 ﺔﺤﯿﻔﺼﻟا ﺔﻟﺎﻄﻋ ﺰﻛﺮﻣ َو IBCD

ﺔﺤﯿﻔﺼﻟا ﺔﻟﺎﻄﻋ ﺰﻛﺮﻣ H ﺔﺤﯿﻔﺼﻟا ﺔﻟﺎﻄﻋ ﺰﻛﺮﻣ ﻲﻟﺎﺘﻟﺎﺑ و AIEF

ABCDEF

ﺔﻟﺎﻄﻋ ﺰﻛﺮﻣ ﻮھ َو O

نذإ H

( )

G∈ OH

(2 ﺮﺒﺘﻌﻧ ﻦﻜﻟ ﺔﻘﯾﺮﻄﻟا ﺲﻔﻨﺑ ﻦﯿﻠﯿﻄﺘﺴﻤﻟا

َو ABCJ نأ ﻦھﺮﺒﻧ JDEF

(

^' ^'

)

G∈ O H ﺺﻠﺨﺘﺳا

ﺔﯿﻧﺎﺜﻟا ﺔﻘﯾﺮﻄﻟا :

( )( )

{

^{, 2} ^,3

}

G = O H ﻞﯿﻄﺘﺴﻤﻟا ﺔﺣﺎﺴﻣ نﻷ

ﻲھ IBCD ﻞﯿﻄﺘﺴﻤﻟا ﺔﺣﺎﺴﻣ َو 2

ﻲھ AIEF 3

ﻢﻠﻌﻤﻟا ﻲﻓ

(

^{A I J}^,^{r ur}^,

)

ﺎﻨﯾﺪﻟ 1 3 :

2 2, H  

 

 

َو 2 ,1 O 2

 

 

ﻲﻟﺎﺘﻟﺎﺑ و 11 11 :

10 10, G 

 

 

: 102 ﺔﻘﯾﺮﻄﻟا : 1

ﺮﺒﺘﻌﻧ O1

ﻞﯿﻄﺘﺴﻤﻟا ﺰﻛﺮﻣ َو ABCI

O2

ﻞﯿﻄﺘﺴﻤﻟا ﺰﻛﺮﻣ ﺔﺤﯿﻔﺼﻟا ﺔﻟﺎﻄﻋ ﺰﻛﺮﻣ و IDEF

ﻮھ

( )( )

{

¹^{, 2} ²^{, 4}

} { ⁽

¹^,1

⁾⁽

²^{, 2}

⁾ }

G = O O = O O ىﺮﺧأ قﺮﻃ كﺎﻨھ و

:: 103 ﺔﻠﻤﺠﻟا ﺢﺟﺮﻣ ﻲﯿﺛاﺪﺣإ ﺐﺴﺤﻧ

( ) ( ) ( )

:

{

^K³⁰⁰ ^{5 ,5 ,}^L⁶⁰⁰ ^{15,5 ,}^J¹⁰⁰ ^10,15

}

104 (2

( )

I ∈ OA نﻷ

: ﻒﺼﺘﻨﻣ ﻮھ A

[

^{G G}¹ ²

]

و G3

ﻰﻟإ ﻲﻤﺘﻨﯾ

( )

^OA

(3 نأ ﻆﺣﻻ

( )( )( )

:

{

¹^{, 36} ²^{, 36} ³^,18

}

I = G G G

(4 14 2 OI = 3