اﻟﺠﺪاء اﻟﺴﻠﻤﻲ و ﺗﻄﺒﻴﻘﺎﺗﻪ ﻓﻲ اﻟﻔﻀﺎء -Iاﻟﺠﺪاء اﻟﺴﻠﻤﻲ -1اﻟﺠﺪاء اﻟﺴﻠﻤﻲ ﻟﻤﺘﺠﻬﺘﻴﻦ ﻣﺴﺘﻘﻴﻤﻴﺘﻴﻦ ﺗﻌﺮﻳﻒ ﻟﺘﻜﻦ vو uﻣﺘﺠﻬﺘﻴﻦ ﻣﺴﺘﻘﻴﻤﻴﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء V3 اﻟﺠﺪاء اﻟﺴﻠﻤﻲ ﻟﻠﻤﺘﺠﻬﺘﻴﻦ vو uهﻮ اﻟﻌﺪد اﻟﺤﻘﻴﻘﻲ اﻟﺬي ﻳﺮﻣﺰ ﻟﻪ ﺑـ u ⋅vو اﻟﻤﻌﺮف آﻤﺎ ﻳﻠﻲ. u ⋅v = u × v * إذا آﺎن ﻟﻠﻤﺘﺠﻬﺘﻴﻦ vو uﻧﻔﺲ اﻟﻤﻨﺤﻰ ﻓﺎن * إذا آﺎن ﻟﻠﻤﺘﺠﻬﺘﻴﻦ vو uﻣﻨﺤﻴﺎن ﻣﺘﻌﺎآﺴﺎن ﻓﺎن u ⋅v = − u × v u ⋅v = 0 ﻓﺎن * إذا آﺎن u = 0أو v = 0 -2اﻟﺠﺪاء اﻟﺴﻠﻤﻲ ﻟﻤﺘﺠﻬﺘﻴﻦ ﻏﻴﺮ ﻣﺴﺘﻘﻴﻤﻴﺘﻴﻦ ﺗﻌﺮﻳﻒ ﻟﺘﻜﻦ vو uﻣﺘﺠﻬﺘﻴﻦ ﻏﻴﺮ ﻣﺴﺘﻘﻴﻤﻴﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء V3و Aﻧﻘﻄﺔ ﻣﻦ اﻟﻔﻀﺎء . E ﺗﻮﺟﺪ ﻧﻘﻄﺘﺎن Bو Cﻣﻦ اﻟﻔﻀﺎء ﺣﻴﺚ u = ABو v = ACاﻟﺠﺪاء اﻟﺴﻠﻤﻲ ﻟﻠﻤﺘﺠﻬﺘﻴﻦ vو uهﻮ اﻟﻌﺪد اﻟﺤﻘﻴﻘﻲ u ⋅vاﻟﻤﻌﺮف آﻤﺎ ﻳﻠﻲ ' u ⋅ v = AB ⋅ AC = AB ⋅ ACﺣﻴﺚ' Cاﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟـ Cﻋﻠﻰ )(AB BAC زاوﻳﺔ ﻣﻨﻔﺮﺟﺔ ' u ⋅ v = AB ⋅ AC = − AB × AC BAC زاوﻳﺔ ﺣﺎدة ' u ⋅ v = AB ⋅ AC = AB × AC -2ﺻﻴﻐﺔ ﻣﺜﻠﺜﻴﺔ ﻟﻠﺠﺪاء اﻟﺴﻠﻤﻲ ﻟﺘﻜﻦ vو uﻣﺘﺠﻬﺘﻴﻦ ﻏﻴﺮ ﻣﻨﻌﺪﻣﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء V3و Aو Bو Cﺛﻼث ﻧﻘﻂ ﻣﻦ اﻟﻔﻀﺎء ﺣﻴﺚ u = AB و v = ACو θﻗﻴﺎس اﻟﺰاوﻳﺔ BAC * إذا آﺎن 0≤ θ ≺ πﻓﺎن BAC زاوﻳﺔ ﺣﺎدة 2 و ﺣﻴﺚ AC ' = AC ⋅ cos θ ' u ⋅ v = AB ⋅ AC = AB × AC ﻓﺎن u ⋅ v = AB × AC cos θ BAC زاوﻳﺔ ﻣﻨﻔﺮﺟﺔ * إذا آﺎن π ≺ θ ≤ πﻓﺎن 2 ' u ⋅ v = AB ⋅ AC = − AB × ACو ﺣﻴﺚ AC ' = AC ⋅ cos (π − θ ) = − AC cos θ و' Cاﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟـ Cﻋﻠﻰ )(AB ﻓﺎن u ⋅ v = AB × AC cos θ π * إذا آﺎن θ = πﻓﺎن AC ' = 0 2 2 ﺧﺎﺻﻴﺔ إذا آﺎﻧﺖ vو uﻣﺘﺠﻬﺘﻴﻦ ﻏﻴﺮ ﻣﻨﻌﺪﻣﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء V3و Aو Bو Cﺛﻼث ﻧﻘﻂ ﻣﻦ اﻟﻔﻀﺎء ﺣﻴﺚ و ﻣﻨﻪ و v = ACو θﻗﻴﺎس اﻟﺰاوﻳﺔ ) ( AB; AC Moustaouli Mohamed u ⋅v = 0 ﻓﺎن إذن u ⋅ v = AB × AC cos u = AB u ⋅ v = AB × AC cos θ 1 http://arabmaths.ift.fr ﻧﺘﻴﺠﺔ إذا آﺎﻧﺖ vو uﻣﺘﺠﻬﺘﻴﻦ ﻏﻴﺮ ﻣﻨﻌﺪﻣﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء Vو θﻗﻴﺎس اﻟﺰاوﻳﺔ ) (u; v 3 ﺧﺎﺻﻴﺔ ﻟﺘﻜﻦ ABو CDﻣﺘﺠﻬﺘﻴﻦ ﻏﻴﺮ ﻣﻨﻌﺪﻣﺘﻴﻦ اﻟﻤﺴﻘﻄﺎن اﻟﻌﻤﻮدﻳﺎن ﻟـ ; C ' AB ⋅ CD = AB ⋅ C ' Dﺣﻴﺚ ' D ' ; C ﻓﺎن u ⋅v = u × v cos θ ﻋﻠﻰ ) ( ABﺑﺎﻟﺘﻮاﻟﻲ. D -3ﺧﺎﺻﻴﺎت أ -ﺗﻌﺎﻣﺪ ﻣﺘﺠﻬﺘﻴﻦ : ﺗﻌﺮﻳﻒ ﻟﺘﻜﻦ vو uﻣﺘﺠﻬﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء .V3 ﺗﻜﻮن vو uﻣﺘﻌﺎﻣﺪﻳﻦ إذا وﻓﻘﻂ إذا آﺎن u ⊥v ﻧﻜﺘﺐ u ⋅v = 0 ﻣﻼﺣﻈﺔ اﻟﻤﺘﺠﻬﺔ 0ﻋﻤﻮدﻳﺔ ﻋﻠﻰ أﻳﺔ ﻣﺘﺠﻬﺔ ﻣﻦ اﻟﻔﻀﺎء V3 ب -ﻣﻨﻈﻢ ﻣﺘﺠﻬﺔ u = AB ﻟﺘﻜﻦ uﻣﺘﺠﻬﺔ ﻏﻴﺮ ﻣﻨﻌﺪﻣﺔ و Aو Bﻧﻘﻄﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء ﺣﻴﺚ u ⋅u 0 إذن ﻟﻜﻞ ﻣﺘﺠﻬﺔ ﻏﻴﺮ ﻣﻨﻌﺪﻣﺔ u 2 اﻟﻌﺪد اﻟﺤﻘﻴﻘﻲ u ⋅ uﻳﺴﻤﻰ اﻟﻤﺮﺑﻊ اﻟﺴﻠﻤﻲ ﻟـ uو ﻳﻜﺘﺐ u اﻟﻌﺪد u 2 ﻳﺴﻤﻰ ﻣﻨﻈﻢ اﻟﻤﺘﺠﻬﺔ uوﻳﻜﺘﺐ u = u2 ﻣﻼﺣﻈﺔ * وﻣﻨﻪ u ⋅ u = AB 2 2 u ج -ﺧﺎﺻﻴﺎت ∈ ∀α ∈ V33 ) ∀ (u , v , w ( u + v ) = u + v + 2u ⋅ v ( u − v )2 = u 2 + v 2 − 2u ⋅ v ( u + v )( u − v ) = u 2 − v 2 2 ﻣﺘﻄﺎﺑﻘﺎت هﺎﻣﺔ 2 2 u ⋅v = v ⋅ u * u ⋅ (v + w ) = u ⋅v + u ⋅w * (v + w ) ⋅ u = v ⋅ u + w ⋅ u * ) u ⋅ αv = αu ⋅v = α × (u ⋅v * -IIﺻﻴــــــﻎ ﺗﺤﻠﻴﻠﻴـــــــــﺔ -1اﻷﺳﺎس و اﻟﻤﻌﻠﻢ اﻟﻤﺘﻌﺎﻣﺪان اﻟﻤﻤﻨﻈﻤﺎن ﺗﻌﺮﻳﻒ ﻟﺘﻜﻦ iو jو kﺛﻼث ﻣﺘﺠﻬﺎت ﻏﻴﺮ ﻣﺴﺘﻮاﺋـــــﻴﺔ ﻣﻦ اﻟﻔﻀﺎء V3و Oﻧﻘﻄﺔ ﻣﻦ اﻟﻔﻀﺎء. ) (i ; j ;kأﺳﺎ س ﻟﻠﻔﻀﺎء V3 ﻳﻜﻮن اﻷﺳﺎس ) (i ; j ;kﻣﺘﻌﺎﻣﺪ )أو اﻟﻤﻌﻠﻢ ) (O ; i ; j ; kﻣﺘﻌﺎﻣﺪ( إذا وﻓﻘﻂ إذا آﺎﻧﺖ اﻟﻤﺘﺠﻬﺎت iو jوk ﻣﺘﻌﺎﻣﺪة ﻣﺜﻨﻰ ﻣﺜﻨﻰ. ﻳﻜﻮن اﻷﺳﺎس ) (i ; j ;kﻣﺘﻌﺎﻣﺪ و ﻣﻤﻨﻈﻢ )أو اﻟﻤﻌﻠﻢ ) (O ; i ; j ; kﺗﻌﺎﻣﺪ وﻣﻤﻨﻈﻢ( إذا وﻓﻘﻂ إذا آﺎﻧﺖ اﻟﻤﺘﺠﻬﺎت iو jو kﻣﺘﻌﺎﻣﺪة ﻣﺜﻨﻰ ﻣﺜﻨﻰ و i = j = k = 1 -2اﻟﺼﻴﻐﺔ اﻟﺘﺤﻠﻴﻠﻴﺔ ﻟﻠﺠﺪاء اﻟﺴﻠﻤﻲ أ -ﺧﺎﺻﻴﺔ اﻟﻔﻀﺎء ﻣﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ.م.م ) (O ; i ; j ; k إذا آﺎﻧﺖ ) u( x; y; zو )' v( x'; y'; zﻓﺎن ' u ⋅ v = xx '+ yy '+ zz ﻣﻼﺣﻈﺔ إذا آﺎﻧﺖ ) u( x; y; zﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻌﻠﻢ.م.م ) (O ; i ; j ; kﻓﺎن u ⋅k = z ; u⋅j =y ; u ⋅i = x ب -اﻟﺼﻴﻐﺔ اﻟﺘﺤﻠﻴﻠﻴﺔ ﻟﻤﻨﻈﻢ ﻣﺘﺠﻬﺔ و ﻟﻤﺴﺎﻓﺔ ﺑﻴﻦ ﻧﻘﻄﺘﻴﻦ * -إذا آﺎﻧﺖ ) u( x; y; zﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻌﻠﻢ.م.م ) (o;i ; j ;kﻓﺎن u = x + y + z 2 Moustaouli Mohamed 2 2 2 http://arabmaths.ift.fr * -اذا آﺎﻧﺖ ) A ( x A ; y A ; z Aو ) B ( xB ; yB ; z Bﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻌﻠﻢ.م.م ) (o;i ; j ;k ﻓﺎن ( xB − x A ) 2 + ( y B − y A ) 2 + ( z B − z A ) 2 = AB ج – ﺗﻌﺎﻣﺪ ﻣﺘﺠﻬﺘﻴﻦ ﺧﺎﺻﻴﺔ ) u( x; y; zو )' v( x'; y'; zﻣﺘﺠﻬﺘﺎن ﻣﻦ ﻓﻀﺎء ﻣﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ.م.م ) (O ; i ; j ; k 'u ⊥v اذا وﻓﻘﻂ اذا آﺎن xx '+ yy '+ zz ' = 0 ﺗﻤﺮﻳﻦ -1ﺣﺪد ﻣﺘﺠﻬﺔ wواﺣﺪﻳﺔ وﻋﻤﻮدﻳﺔ ﻋﻠﻰ ) u(−1;1;1و ) v(1;−2;0 - 2ﺣﺪد ﻣﺘﺠﻬﺔ wﻋﻤﻮدﻳﺔ ﻋﻠﻰ ) u(1;1;0و ) v(0;2;1و 3 ﺗﻤﺮﻳﻦ ﻧﻌﺘﺒﺮ ) ( ) A 1;1; 2و 2; − 2;0 ( B ) = w ( و C −1; −1; − 2 ﺑﻴﻦ أن ABCﻣﺜﻠﺚ ﻣﺘﺴﺎوي اﻟﺴﺎﻗﻴﻦ وﻗﺎﺋﻢ اﻟﺰاوﻳﺔ Moustaouli Mohamed 3 http://arabmaths.ift.fr