اﻟﺠﺪاء اﻟﺴﻠﻤﻲ ﻓﻲ اﻟﻔﻀﺎء و ﺗﻄﺒﻴﻘﺎﺗﻪ -Iاﻟﺠﺪاء اﻟﺴﻠﻤﻲ -1ﺗﻌﺮﻳﻒ ﻟﺘﻜﻦ vو uﻣﺘﺠﻬﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء ،و Aو Bو Cﻧﻘﻂ ﻣﻦ اﻟﻔﻀﺎء ﺣﻴﺚ u = ABو . v = AC ﻳﻮﺟﺪ ﻋﻠﻰ اﻻﻗﻞ ﻣﺴﺘﻮى ) ( Pﺿﻤﻦ اﻟﻔﻀﺎء ﻳﻤﺮ ﻣﻦ اﻟﻨﻘﻂ Aو Bو . C اﻟﺠﺪاء اﻟﺴﻠﻤﻲ ﻟﻠﻤﺘﺠﻬﺘﻴﻦ vو uﻓﻲ اﻟﻔﻀﺎء هﻮ اﻟﺠﺪاء اﻟﺴﻠﻤﻲ AB ⋅ ACﻓﻲ اﻟﻤﺴﺘﻮى ) ( Pﻧﺮﻣﺰ ﻟﻪ ﺑـ u ⋅ v
ﻣﻠﺤﻮﻇﺔ ﺟﻤﻴﻊ ﺧﺎﺻﻴﺎت اﻟﺠﺪاء اﻟﺴﻠﻤﻲ ﻓﻲ اﻟﻤﺴﺘﻮى ﺗﻤﺪد إﻟﻰ اﻟﻔﻀﺎء -2ﻧﺘﺎﺋﺞ ﻟﺘﻜﻦ vو uﻣﺘﺠﻬﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء ،و Aو Bو Cﻧﻘﻂ ﻣﻦ اﻟﻔﻀﺎء ﺣﻴﺚ u = ABو . v = AC * إذا آﺎن v ≠ 0و u ≠ 0ﻓﺎن u ⋅ v = AB × AC × cos BAC u ⋅v = 0 ﻓﺎن * إذا آﺎن u = 0أو v = 0 * إذا آﺎن u ≠ 0ﻓﺎن ' u ⋅ v = AB ⋅ AC = AB ⋅ AC ﺣﻴﺚ' Cاﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟـ Cﻋﻠﻰ )(AB 1 = u ⋅v * AB 2 + AC 2 − BC 2 2
(
)
-3ﻣﻨﻈﻢ ﻣﺘﺠﻬﺔ ﻟﺘﻜﻦ uﻣﺘﺠﻬﺔ و Aو Bﻧﻘﻄﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء ﺣﻴﺚ
u = AB
اﻟﻌﺪد اﻟﺤﻘﻴﻘﻲ u ⋅ uﻳﺴﻤﻰ اﻟﻤﺮﺑﻊ اﻟﺴﻠﻤﻲ ﻟـ uو ﻳﻜﺘﺐ u 2 = AB 2 اﻟﻌﺪد اﻟﺤﻘﻴﻘﻲ اﻟﻤﻮﺟﺐ u 2
ﻳﺴﻤﻰ ﻣﻨﻈﻢ اﻟﻤﺘﺠﻬﺔ uﻧﻜﺘﺐ u = u 2
ﻣﻼﺣﻈﺔ و آﺘﺎﺑﺔ 2 * u = u2 *
إذا آﺎن
v ≠ 0وu ≠ 0
-4ﺧﺎﺻﻴﺎت ∈ ∀α
3 3
ﻓﺎن
) (
u ⋅ v = u v cos u ; v
∀ (u ,v ,w ) ∈V
ﻣﺘﻄﺎﺑﻘﺎت هﺎﻣﺔ
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 -5ﺗﻌﺎﻣﺪ ﻣﺘﺠﻬﺘﻴﻦ : ﺗﻌﺮﻳﻒ ﻟﺘﻜﻦ vو uﻣﺘﺠﻬﺘﻴﻦ ﻣﻦ اﻟﻔﻀﺎء .V3
ﺗﻜﻮن vو uﻣﺘﻌﺎﻣﺪﻳﻦ إذا وﻓﻘﻂ إذا آﺎن
u ⋅v = 0
ﻣﻼﺣﻈﺔ اﻟﻤﺘﺠﻬﺔ 0ﻋﻤﻮدﻳﺔ ﻋﻠﻰ أﻳﺔ ﻣﺘﺠﻬﺔ ﻣﻦ اﻟﻔﻀﺎء V3 ﺗﻤﺮﻳﻦ اﻟﻤﻜﻌﺐ ABCDEFGHاﻟﺬي ﻃﻮل ﺣﺮﻓﻪ a أﺣﺴﺐ AE.BGو AE.AGو AG.EB
1
ﻧﻜﺘﺐ
(u + v ) = u + v + 2u ⋅v 2 (u − v ) = u 2 + v 2 − 2u ⋅v (u + v )(u − v ) = u 2 − v 2
u ⊥v
2
2
2