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20 Anwendungen

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20.3 Vektorielle Darstellung der Ebene

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20.3.2 Drei-Punkte-Form

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Beispiel 20 - 212
Umwandlung der Ebenendarstellung: Drei-Punkte-Form in eine Parameterdarstellung .
Gegeben sind die drei Punkte mit den Ortsvektoren

r 1 =( 1 5 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Waa8HaaeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaaGccaGLxdcacqGH 9aqpdaqadaqaauaabeqadeaaaeaacaaIXaaabaGaaGynaaqaaiaaic daaaaacaGLOaGaayzkaaaaaa@3E75@  ,     r 2 =( 2 1 8 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Waa8HaaeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaaGccaGLxdcacqGH 9aqpdaqadaqaauaabeqadeaaaeaacqGHsislcaaIYaaabaGaeyOeI0 IaaGymaaqaaiaaiIdaaaaacaGLOaGaayzkaaaaaa@4055@ und  r 3 =( 2 0 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Waa8HaaeaacaWGYbWaaSbaaSqaaiaaiodaaeqaaaGccaGLxdcacqGH 9aqpdaqadaqaauaabeqadeaaaeaacaaIYaaabaGaaGimaaqaaiaaig daaaaacaGLOaGaayzkaaaaaa@3E74@ . .
Die Parameterdarstellung der Ebene lautet dann .

r (P)=( x 1 y 1 z 1 )+λ( x 2 x 1 y 2 y 1 z 2 z 1 )+μ( x 3 x 1 y 3 y 1 z 3 z 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Waa8HaaeaacaWGYbaacaGLxdcacaGGOaGaamiuaiaacMcacqGH9aqp daqadaqaauaabeqadeaaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaa GcbaGaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaadQhadaWgaaWc baGaaGymaaqabaaaaaGccaGLOaGaayzkaaGaey4kaSIaeq4UdW2aae WaaeaafaqabeWabaaabaGaamiEamaaBaaaleaacaaIYaaabeaakiab gkHiTiaadIhadaWgaaWcbaGaaGymaaqabaaakeaacaWG5bWaaSbaaS qaaiaaikdaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaaIXaaabeaa aOqaaiaadQhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG6bWaaS baaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaiabgUcaRiabeY7a TnaabmaabaqbaeqabmqaaaqaaiaadIhadaWgaaWcbaGaaG4maaqaba GccqGHsislcaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyEamaa BaaaleaacaaIZaaabeaakiabgkHiTiaadMhadaWgaaWcbaGaaGymaa qabaaakeaacaWG6bWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaamOE amaaBaaaleaacaaIXaaabeaaaaaakiaawIcacaGLPaaaaaa@6874@ =.
1 5 0 +λ -2 - 1 -1 - 5 8 - 0 +μ 2 - 1 0 - 5 1 - 0 .
= 1 5 0 +λ -3 -6 8 +μ 1 -5 1 .

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