# espace by AbdrrahimEddafi

VIEWS: 35 PAGES: 2

• pg 1
```									    ‫)ﻣﺤﻤﺪ اﻟﻜﯿﺎل(‬                  ‫اﻟﮫﻨﺪﺳﺔ اﻟﻔﻀﺎﺋﯿﺔ‬
‫‪í‬‬
‫‪rr r‬‬
‫(‬           ‫)‬
‫ﻓﻲ ﺳﯿﺎق ھﺬا اﻟﻤﻠﺨﺺ ﻟﯿﻜﻦ اﻟﻔﻀﺎء ﻣﻨﺴﻮﺑﺎ إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ‪o, i, j,k‬‬

‫‪Ë‬اﻟﺼﯿﻐﺔ اﻟﺘﺤﻠﯿﻠﯿﺔ ل: اﻟﺠﺪاء اﻟﺴﻠﻤﻲ- ﻣﻨﻈﻢ ﻣﺘﺠﮫﺔ- اﻟﺠﺪاء اﻟﻤﺘﺠﮫﻲ:‬
‫‪r‬‬                  ‫‪r‬‬
‫ﻟﺘﻜﻦ ) ‪ u ( a,b,c‬و )'‪ v ( a ', b ',c‬ﻣﺘﺠﮫﺘﯿﻦ ﻣﻦ 3‪J‬‬
‫‪rr‬‬
‫'‪u.v = aa '+ bb'+ cc‬‬                                       ‫·‬
‫‪r‬‬
‫²‪u = a² + b² + c‬‬                                              ‫·‬
‫‪r‬‬
‫'‪i a a‬‬
‫‪r r r‬‬            ‫‪b b' r a a ' r a a ' r‬‬
‫= '‪u Ù v = j b b‬‬      ‫-‪i‬‬      ‫+‪j‬‬      ‫‪k‬‬                       ‫·‬
‫‪r‬‬        ‫'‪c c‬‬    ‫'‪c c‬‬    ‫'‪b b‬‬
‫'‪k c c‬‬

‫‪Ë‬اﻟﻤﺴﺎﻓﺔ:‬
‫اﻟﻤﺴﺎﻓﺔ ﺑﯿﻦ ﻧﻘﻄﺘﯿﻦ ‪ A‬و ‪ B‬ھﻲ:‬
‫‪r‬‬
‫‪uuu‬‬
‫= ‪AB = AB‬‬         ‫² ) ‪( x B - x A ) ² + ( yB - yA ) ² + ( z B - z A‬‬
‫ﻣﻌﺎدﻟﺘﻪ اﻟﺪﻳﻜﺎرﺗﯿﺔ: 0 = ‪ ax + by + cz + d‬ھﻲ:‬             ‫و ﻣﺴﺘﻮى ) ‪( P‬‬   ‫اﻟﻤﺴﺎﻓﺔ ﺑﯿﻦ ﻧﻘﻄﺔ ‪M‬‬
‫‪ax M + by M + cz M + d‬‬
‫= ) ) ‪d ( M, ( R‬‬
‫²‪a² + b² + c‬‬
‫‪r‬‬
‫اﻟﻤﺴﺎﻓﺔ ﺑﯿﻦ ﻧﻘﻄﺔ ‪ M‬و ﻣﺴﺘﻘﯿﻢ ) ‪ D ( A, u‬ھﻲ:‬
‫‪r‬‬
‫‪uuuu r‬‬
‫‪AM Ù u‬‬
‫= ) ) ‪d ( A, ( D‬‬      ‫‪r‬‬
‫‪u‬‬

‫‪Ë‬ﻣﻌﺎدﻟﺔ ﻣﺴﺘﻮى:‬
‫‪r‬‬
‫)‪(R‬‬   ‫0 = ‪ n ( a, b,c ) Û ( R ) : ax + by + cz + d‬ﻣﺘﺠﮫﺔ ﻣﻨﻈﻤﯿﺔ ﻋﻠﻰ اﻟﻤﺴﺘﻮى‬
‫‪r‬‬   ‫‪r‬‬
‫‪uuu uuu‬‬
‫(‬ ‫إذا ﻛﺎﻧﺖ ‪ A‬و ‪ B‬و ‪ C‬ﻧﻘﻂ ﻏﯿﺮ ﻣﺴﺘﻘﯿﻤﺔ ﻓﺈن ‪ AB Ù AC‬ﻣﺘﺠﮫﺔ ﻣﻨﻈﻤﯿﺔ ﻋﻠﻰ اﻟﻤﺴﺘﻮى ) ‪ABC‬‬
‫و ﻓﻲ ھﺬه اﻟﺤﺎﻟﺔ ﻳﻤﻜﻦ ﺗﺤﺪﻳﺪ ﻣﻌﺎدﻟﺔ اﻟﻤﺴﺘﻮى ) ‪ ( ABC‬ﺑﺎﻻﺳﺘﻌﺎﻧﺔ ﺑﺎﻟﺘﻜﺎﻓﺆ اﻟﺘﺎﻟﻲ :‬
‫‪r‬‬   ‫‪r‬‬   ‫‪r‬‬
‫‪uuuu uuu uuu‬‬
‫(‬
‫0 = ‪M Î ( ABC ) Û AM. AB Ù AC‬‬           ‫)‬
‫‪Ë‬ﻣﻌﺎدﻟﺔ ﻓﻠﻜﺔ:‬

‫ﻣﻌﺎدﻟﺔ ﻓﻠﻜﺔ ﻣﺮﻛﺰھﺎ ) ‪ W ( a, b,c‬و ﺷﻌﺎﻋﮫﺎ ‪ R‬ھﻲ:‬
‫²‪( x - a ) ² + ( y - b ) ² + ( z - c ) ² = R‬‬

‫23‬
‫ﻣﻌﺎدﻟﺔ ﻓﻠﻜﺔ ) ‪ ( S‬أﺣﺪ أﻗﻄﺎرھﺎ ]‪ [ AB‬ﻳﻤﻜﻦ ﺗﺤﺪﻳﺪھﺎ‬
‫‪r‬‬
‫‪uuuu uuur‬‬
‫0 = ‪M Î ( S) Û AM . BM‬‬        ‫ﺑﺎﻻﺳﺘﻌﺎﻧﺔ ﺑﺎﻟﺘﻜﺎﻓﺆ اﻟﺘﺎﻟﻲ:‬

‫] [‬                  ‫) (‬
‫‪AB‬‬
‫ﻣﻼﺣﻈﺔ:اﻟﻔﻠﻜﺔ ‪ S‬ﻣﺮﻛﺰھﺎ ‪ W‬ﻣﻨﺘﺼﻒ ‪ AB‬و ﺷﻌﺎﻋﮫﺎ‬
‫2‬

‫‪Ë‬ﺗﻘﺎﻃﻊ ﻓﻠﻜﺔ ) ‪ S ( W, R‬و ﻣﺴﺘﻮى 0 = ‪( R ) : ax + by + cz + d‬‬
‫ﻟﺘﻜﻦ ‪ H‬اﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟﻠﻤﺮﻛﺰ ‪ W‬ﻋﻠﻰ اﻟﻤﺴﺘﻮى ) ‪( R‬‬
‫ﻧﻀﻊ: ) ) ‪d = WH = d ( W; ( R‬‬

‫‪ R‬ﻳﻘﻄﻊ‬‫اﻟﻤﺴﺘﻮى ) (‬
‫) (‬
‫اﻟﻤﺴﺘﻮى ‪ R‬ﻣﻤﺎس‬
‫اﻟﻤﺴﺘﻮى ) ‪( R‬‬
‫اﻟﻔﻠﻜﺔ )‪ (S‬وﻓﻖ داﺋﺮة ) ‪( C‬‬
‫ﻣﺮﻛﺰھﺎ: ‪H‬‬
‫ﻟﻠﻔﻠﻜﺔ )‪(S‬‬                          ‫ﻻ ﻳﻘﻄﻊ اﻟﻔﻠﻜﺔ )‪(S‬‬
‫ﻓﻲ اﻟﻨﻘﻄﺔ ‪H‬‬
‫ﺷﻌﺎﻋﮫﺎ: ²‪r = R ² - d‬‬
‫‪Ë‬ﺗﻘﺎﻃﻊ ﻓﻠﻜﺔ ) ‪ S ( W, R‬و ﻣﺴﺘﻘﯿﻢ ) ‪: ( D‬‬
‫ﻟﺘﻜﻦ ‪ H‬اﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟﻠﻤﺮﻛﺰ ‪ W‬ﻋﻠﻰ اﻟﻤﺴﺘﻘﯿﻢ ) ‪( D‬‬
‫ﻧﻀﻊ: ) ) ‪d = WH = d ( W; ( D‬‬

‫) ‪ ( D‬ﻳﺨﺘﺮق اﻟﻔﻠﻜﺔ )‪(S‬‬   ‫اﻟﻤﺴﺘﻘﯿﻢ‬           ‫‪ D‬ﻣﻤﺎس‬‫) (‬    ‫اﻟﻤﺴﺘﻘﯿﻢ‬                     ‫اﻟﻤﺴﺘﻮى ) (‬
‫‪ R‬و اﻟﻔﻠﻜﺔ‬

‫ﻓﻲ ﻧﻘﻄﺘﯿﻦ ﻣﺨﺘﻠﻔﺘﯿﻦ‬                   ‫ﻟﻠﻔﻠﻜﺔ )‪ ( S‬ﻓﻲ اﻟﻨﻘﻄﺔ ‪H‬‬                 ‫)‪ ( S‬ﻻ ﻳﺘﻘﺎﻃﻌﺎن‬

‫33‬

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