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首页 空间解析几何课件精讲

空间解析几何课件精讲.pdf

空间解析几何课件精讲

Vicky
2013-12-29 0人阅读 举报 0 0 暂无简介

简介:本文档为《空间解析几何课件精讲pdf》,可适用于高等教育领域

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|sin∠(a,b),ِ•†a,b©OR†§…{a,b,a×b}�¤mÃXw,kµa×=×a=a×a=a×(−a)=˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚È�½ÂÚL«mÃX††ÃX(�˜‡º¡‡�)µe{a,b,c}´mÃX§�ѤkmÃXچÃX|Ü"mµ{a,b,c},{b,c,a},{c,a,b}†µ{a,c,b},{b,a,c},{c,b,a}È�½Âµ½Âa†b�È´˜‡•þ§Pa×b,ÙÝ|a×b|=|a||b|sin∠(a,b),ِ•†a,b©OR†§…{a,b,a×b}�¤mÃXw,kµa×=×a=a×a=a×(−a)=˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚È�Ÿ½na∥b⇔a×b=$ŽÆµ{‡�†Æ:a×b=−b×a(ÜÆ:(λa)×b=λ(a×b)©�Æ:a×(bc)=a×ba×cgggµµµþã(ÜÆ´†ê¦(ܧ†g��(ÜƴĤáº=(a×b)×c====a×(b×c)˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚È�Ÿ½na∥b⇔a×b=$ŽÆµ{‡�†Æ:a×b=−b×a(ÜÆ:(λa)×b=λ(a×b)©�Æ:a×(bc)=a×ba×cgggµµµþã(ÜÆ´†ê¦(ܧ†g��(ÜƴĤáº=(a×b)×c====a×(b×c)˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚È�Ÿ^Ȧ†ü•þÑR†�•þ"=ec⊥a,b,Ù¥a�b,Kkc=λ(a×b):�‚�ål(ùp"˜‡ã)d=|AB|×|AP||AB|˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚È�Ÿ^Ȧ†ü•þÑR†�•þ"=ec⊥a,b,Ù¥a�b,Kkc=λ(a×b):�‚�ål(ùp"˜‡ã)d=|AB|×|AP||AB|˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚~KŠ’~("ã)y²µ²o>�¡È´±Ùé�‚>�,˜‡²o>¡È�˜Œ"~y²n���u½n"~y²·‚�sincosúªµ(a×b)=ab−(a·b)¿ddíÑ°Ô(Heron)úª£"㤵SABC=√s(s−a)(s−b)(s−c),Ù¥s=(abc)Š’µSKP!(hint:GAGBGC=)K˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚~KŠ’~("ã)y²µ²o>�¡È´±Ùé�‚>�,˜‡²o>¡È�˜Œ"~y²n���u½n"~y²·‚�sincosúªµ(a×b)=ab−(a·b)¿ddíÑ°Ô(Heron)úª£"㤵SABC=√s(s−a)(s−b)(s−c),Ù¥s=(abc)Š’µSKP!(hint:GAGBGC=)K˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚~KŠ’~("ã)y²µ²o>�¡È´±Ùé�‚>�,˜‡²o>¡È�˜Œ"~y²n���u½n"~y²·‚�sincosúªµ(a×b)=ab−(a·b)¿ddíÑ°Ô(Heron)úª£"㤵SABC=√s(s−a)(s−b)(s−c),Ù¥s=(abc)Š’µSKP!(hint:GAGBGC=)K˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚~KŠ’~("ã)y²µ²o>�¡È´±Ùé�‚>�,˜‡²o>¡È�˜Œ"~y²n���u½n"~y²·‚�sincosúªµ(a×b)=ab−(a·b)¿ddíÑ°Ô(Heron)úª£"㤵SABC=√s(s−a)(s−b)(s−c),Ù¥s=(abc)Š’µSKP!(hint:GAGBGC=)K˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚•þ�·ÜȽÂn‡•þ�·ÜÈ(a,b,c)Def=====(a×b)×c´˜‡êþ§ÙAۿ´n•þ|¤�²¡N�k•Nȧ=µ(a,b,c)=|a×b||c|cosθd½ÂŒ±wѵ(a,b,c)>⇐⇒{a,b,c}mÃX(a,b,c)<⇐⇒{a,b,c}†ÃXo�u"Q§·‚ke¡�½nµ˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚•þ�·ÜȽÂn‡•þ�·ÜÈ(a,b,c)Def=====(a×b)×c´˜‡êþ§ÙAۿ´n•þ|¤�²¡N�k•Nȧ=µ(a,b,c)=|a×b||c|cosθd½ÂŒ±wѵ(a,b,c)>⇐⇒{a,b,c}mÃX(a,b,c)<⇐⇒{a,b,c}†ÃXo�u"Q§·‚ke¡�½nµ˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚·ÜÈ�Ÿ½na,b,c�¡⇐⇒(a,b,c)=�†{K(ÙP¬^)µ(a,b,c)=(b,c,a)=(c,a,b)=−(a,c,b)=−(b,a,c)=−(c,b,a)n­‚Ÿ(ùpù˜eŸo´õ­‚)µ(aa,b,c)=(a,b,c)(a,b,c)(λa,b,c)=λ(a,b,c)gggµµµÐm(λaλa,µbµb,νcνc),kõ�‘º˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚·ÜÈ�Ÿ½na,b,c�¡⇐⇒(a,b,c)=�†{K(ÙP¬^)µ(a,b,c)=(b,c,a)=(c,a,b)=−(a,c,b)=−(b,a,c)=−(c,b,a)n­‚Ÿ(ùpù˜eŸo´õ­‚)µ(aa,b,c)=(a,b,c)(a,b,c)(λa,b,c)=λ(a,b,c)gggµµµÐm(λaλa,µbµb,νcνc),kõ�‘º˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚·ÜÈ�Ÿ½na,b,c�¡⇐⇒(a,b,c)=�†{K(ÙP¬^)µ(a,b,c)=(b,c,a)=(c,a,b)=−(a,c,b)=−(b,a,c)=−(c,b,a)n­‚Ÿ(ùpù˜eŸo´õ­‚)µ(aa,b,c)=(a,b,c)(a,b,c)(λa,b,c)=λ(a,b,c)gggµµµÐm(λaλa,µbµb,νcνc),kõ�‘º˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚·ÜÈ�Ÿ½na,b,c�¡⇐⇒(a,b,c)=�†{K(ÙP¬^)µ(a,b,c)=(b,c,a)=(c,a,b)=−(a,c,b)=−(b,a,c)=−(c,b,a)n­‚Ÿ(ùpù˜eŸo´õ­‚)µ(aa,b,c)=(a,b,c)(a,b,c)(λa,b,c)=λ(a,b,c)gggµµµÐm(λaλa,µbµb,νcνc),kõ�‘º˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚˜‡~f~a,b,cØ�¡"ò¿•þd�¤a,b,c�‚|ܧ=¦¢êx,y,z,¦�d=xaybzc)))µµµü>�Xe·Üȵ(d,b,c)=(xaybzc,b,c)=x(a,b,c)y(b,b,c)z(c,b,c)=x(a,b,c)�x=(d,b,c)(a,b,c)˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚˜‡~f~a,b,cØ�¡"ò¿•þd�¤a,b,c�‚|ܧ=¦¢êx,y,z,¦�d=xaybzc)))µµµü>�Xe·Üȵ(d,b,c)=(xaybzc,b,c)=x(a,b,c)y(b,b,c)z(c,b,c)=x(a,b,c)�x=(d,b,c)(a,b,c)˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚˜‡~fÓnŒ¦�y,z(J�e¡µx=(d,b,c)(a,b,c),y=(a,d,c)(a,b,c),z=(a,b,d)(a,b,c)ùҴͶ�Ž{K(Cramer’sRule),§`²µÚn˜m˜•þ'un‡Ø�¡•þ�©)ª´˜�"ù‡(Ø´½Â“‹Iù˜Vg�Ä:"˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚•þ�V­È(a×b)×c¡V­È§ÙÐmªXeµÚn(a×b)×a=(a)b−(a·b)ay²Ñ§–Ö¿"½n(a×b)×c=(a·c)b−(b·c)ay²Ñ§–Ö¿"Úna×(b×c)=−(b×c)×a=(a·c)b−(a·b)c˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚•þ�V­È(a×b)×c¡V­È§ÙÐmªXeµÚn(a×b)×a=(a)b−(a·b)ay²Ñ§–Ö¿"½n(a×b)×c=(a·c)b−(b·c)ay²Ñ§–Ö¿"Úna×(b×c)=−(b×c)×a=(a·c)b−(a·b)c˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚•þ�V­È(a×b)×c¡V­È§ÙÐmªXeµÚn(a×b)×a=(a)b−(a·b)ay²Ñ§–Ö¿"½n(a×b)×c=(a·c)b−(b·c)ay²Ñ§–Ö¿"Úna×(b×c)=−(b×c)×a=(a·c)b−(a·b)c˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚•þ�V­ÈÚn(a×b)×(c×d)=(a,b,d)c−(a,b,c)d=(a,c,d)b−(b,c,d)a()y²Ñ§–Ö¿"~y²äŒ'(Jacobi)ð�ªµ(a×b)×c(b×c)×a(c×a)×b=()yyy²²²µµµÑÐmg,Œ�"˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚•þ�V­ÈÚn(a×b)×(c×d)=(a,b,d)c−(a,b,c)d=(a,c,d)b−(b,c,d)a()y²Ñ§–Ö¿"~y²äŒ'(Jacobi)ð�ªµ(a×b)×c(b×c)×a(c×a)×b=()yyy²²²µµµÑÐmg,Œ�"˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚~KŠ’~y²‚KF(Lagrange)ð�ªµ(a×b)·(c×d)=(a·c)(b·d)−(b·c)(a·d)()yyy²²²µµµU˜)ÒÐm=Œ"Ù¢·‚®²�>L‚KFð�ª�˜‡AϜ))úª()¥c=a,d=b,=��·‚�sincosúªµ(a×b)=ab−(a·b)Š’µSKP!K,ÁXy²�‘���½n(�gŠ’ØÂ)"˜m)ÛAۚa¹có•þ“ê•þق$Ž•þ�Sȕþ�È·ÜÈÚV­È²¡††‚~KŠ’~y²‚KF(Lagrange)ð�ªµ(a×b)·(c×d)=(a·c)(b·d)−(b·c)(a·d)()yyy²²²µµµU˜)ÒÐm=Œ"Ù¢·‚®²�>L‚KFð�ª�˜‡AϜ))úª()¥c=a,d=b,=��·‚�sincosúªµ(a×b)=ab−(a·b)Š’µSKP!K,ÁXy²�‘���½n(�gŠ’ØÂ)"˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IXڕ�‹IX½Â�i,j,k´±Oå:�n‡ü •þ§üüR†§=µ|i|=|j|=|k|=,i·j=i·k=j·k=K{Oi,j,k}�¤˜‡†�‹IX(†�Ie)"½Âþã½Â¥�KR†^‡§‡¦n‡•þØ�¡§K�¤•�‹IX(•�Ie)Note:•�(affine)"fain½"æfain:ù‡cêÆ¥k‚!²�¿g§^uXµ•�‹I!•�C†!•�|ܘm)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IXڕ�‹IX½Â�i,j,k´±Oå:�n‡ü •þ§üüR†§=µ|i|=|j|=|k|=,i·j=i·k=j·k=K{Oi,j,k}�¤˜‡†�‹IX(†�Ie)"½Âþã½Â¥�KR†^‡§‡¦n‡•þØ�¡§K�¤•�‹IX(•�Ie)Note:•�(affine)"fain½"æfain:ù‡cêÆ¥k‚!²�¿g§^uXµ•�‹I!•�C†!•�|ܘm)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IXڕ�‹IX½n�{Oi,j,k}´˜‡•�(†�)‹IX§K¿•þvŒ±˜L«‚|ܵv=xiyjzk,¡(x,y,z)•þvT‹IX¥�‹I§Pv=(x,y,z)yyy²²²µµµ=Ž{K",§˜k˜‡äkÊH�y²§„�á"½Â:�‹IҽT:•»(l�:Ñu�T:�•þ)�‹I"˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IXڕ�‹IX½n�{Oi,j,k}´˜‡•�(†�)‹IX§K¿•þvŒ±˜L«‚|ܵv=xiyjzk,¡(x,y,z)•þvT‹IX¥�‹I§Pv=(x,y,z)yyy²²²µµµ=Ž{K",§˜k˜‡äkÊH�y²§„�á"½Â:�‹IҽT:•»(l�:Ñu�T:�•þ)�‹I"˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IX¥�•þOŽ‚$Žµa=(a,a,a),b=(b,b,b),Kab=(ab,ab,ab),()a−b=(a−b,a−b,a−b),()λa=(λa,λa,λa)()Sȵa·b=abiabjabk=ababab()|a|=√a=√aaa()cos∠(a,b)=ababab√aaa√bbb()˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IX¥�•þOŽ‚$Žµa=(a,a,a),b=(b,b,b),Kab=(ab,ab,ab),()a−b=(a−b,a−b,a−b),()λa=(λa,λa,λa)()Sȵa·b=abiabjabk=ababab()|a|=√a=√aaa()cos∠(a,b)=ababab√aaa√bbb()˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IX¥�•þOŽIIȵi×j=k,j×k=i,k×i=j,j×i=−k,k×j=−i,i×k=−j,a×b=(ab−ab)(i×j)(ab−ab)(i×k)(ab−ab)(j×k)=(ab−ab)i(ab−ab)j(ab−ab)k()�ªªBuPÁµa×b=∣∣∣∣∣∣∣∣∣ijkaaabbb∣∣∣∣∣∣∣∣∣˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål†�‹IX¥�•þOŽIII·Üȵa,bXc§c=(c,c,c),Kµ(a,b,c)=(a×b)·c=∣∣∣∣∣∣aabb∣∣∣∣∣∣c∣∣∣∣∣∣aabb∣∣∣∣∣∣c∣∣∣∣∣∣aabb∣∣∣∣∣∣c=∣∣∣∣∣∣∣∣∣aaabbbccc∣∣∣∣∣∣∣∣∣()˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ålålúªÚ½'©:úªålúªµPP=(x−x,y−y,z−z)|PP|=√(x−x)(y−y)(z−z)()½'©:úªµ†�†¤©þ=Œ"˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ålùÇ~K~®A(x,y,z),B(x,y,z),C(x,y,z),¦ABC�¡È"~®A(x,y,z),B(x,y,z),C(x,y,z),D(x,y,z),¦o¡NABCD�NÈ"~^‹I{y²V­Èúª"‘�Š’µSKP!K(Ù¥K�¯(ØI‰U½�½§Ä©"œ¹)˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ålùÇ~K~®A(x,y,z),B(x,y,z),C(x,y,z),¦ABC�¡È"~®A(x,y,z),B(x,y,z),C(x,y,z),D(x,y,z),¦o¡NABCD�NÈ"~^‹I{y²V­Èúª"‘�Š’µSKP!K(Ù¥K�¯(ØI‰U½�½§Ä©"œ¹)˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ålùÇ~K~®A(x,y,z),B(x,y,z),C(x,y,z),¦ABC�¡È"~®A(x,y,z),B(x,y,z),C(x,y,z),D(x,y,z),¦o¡NABCD�NÈ"~^‹I{y²V­Èúª"‘�Š’µSKP!K(Ù¥K�¯(ØI‰U½�½§Ä©"œ¹)˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ålùÇ~K~®A(x,y,z),B(x,y,z),C(x,y,z),¦ABC�¡È"~®A(x,y,z),B(x,y,z),C(x,y,z),D(x,y,z),¦o¡NABCD�NÈ"~^‹I{y²V­Èúª"‘�Š’µSKP!K(Ù¥K�¯(ØI‰U½�½§Ä©"œ¹)˜m)ÛAۚa¹có•þ“겡††‚‹IX¥�•þOŽ²¡�§˜m†‚§²¡††‚�k'¯K!ål:{ª§®²¡Ω,:P(x,y,z)∈Ω,•þN⊥Ω,¡{•þ({•þ˜cêÆ¥˜„‡¦´ü •þ)ã:˜m²¡Kéu²¡þ¿˜:P(x,y,z)∈

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