espace by AbdrrahimEddafi

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									    ‫)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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