Telechargé par abdelaziz hmidat

# biaz101

publicité
```‫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 &times; v‬‬
‫* إذا آﺎن ﻟﻠﻤﺘﺠﻬﺘﻴﻦ ‪ v‬و ‪ u‬ﻧﻔﺲ اﻟﻤﻨﺤﻰ ﻓﺎن‬
‫* إذا آﺎن ﻟﻠﻤﺘﺠﻬﺘﻴﻦ ‪ v‬و ‪ u‬ﻣﻨﺤﻴﺎن ﻣﺘﻌﺎآﺴﺎن ﻓﺎن‬
‫‪u ⋅v = − u &times; 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 &times; AC‬‬
‫‪  BAC ‬زاوﻳﺔ ﺣﺎدة‬
‫‪‬‬
‫‪‬‬
‫' ‪u ⋅ v = AB ⋅ AC = AB &times; 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 &times; AC‬‬
‫ﻓﺎن ‪u ⋅ v = AB &times; AC cos θ‬‬
‫‪  BAC ‬زاوﻳﺔ ﻣﻨﻔﺮﺟﺔ‬
‫* إذا آﺎن ‪ π ≺ θ ≤ π‬ﻓﺎن‬
‫‪‬‬
‫‪‬‬
‫‪2‬‬
‫' ‪ u ⋅ v = AB ⋅ AC = − AB &times; AC‬و ﺣﻴﺚ ‪AC ' = AC ⋅ cos (π − θ ) = − AC cos θ‬‬
‫و'‪ C‬اﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟـ ‪ C‬ﻋﻠﻰ )‪(AB‬‬
‫ﻓﺎن ‪u ⋅ v = AB &times; 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 &times; AC cos‬‬
‫‪u = AB‬‬
‫‪u ⋅ v = AB &times; 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 &times; 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 &times; (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
```