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0/19

20 Anwendungen

0/19/2

20.2 Abstände/Schnittpunkte von Geraden

0/19/2/6

20.2.4 2 Geraden g1 und g2 sind windschief

Vorbemerkung: 0/19/2/6/5 .
Beispiel 20 - 205
Gegeben seien die Geraden: 

g 1 :  x 1 → =( 1 2 0 )+ λ 1 ( 1 1 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4zamaaBaaaleaacaaIXaaabeaakiaacQdacaaMc8+aa8Haaeaa caWG4bWaaSbaaSqaaiaaigdaaeqaaaGccaGLxdcacqGH9aqpdaqada qaauaabeqadeaaaeaacaaIXaaabaGaaGOmaaqaaiaaicdaaaaacaGL OaGaayzkaaGaey4kaSIaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOWaae WaaeaafaqabeWabaaabaGaaGymaaqaaiaaigdaaeaacaaIXaaaaaGa ayjkaiaawMcaaaaa@49EE@ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4zaiaaigdacaGG6aGaaGPaVpaaFiaabaGaamiEamaaBaaaleaa caaIXaaabeaaaOGaay51GaGaeyypa0ZaaeWaaeaafaqabeWabaaaba GaaGymaaqaaiaaikdaaeaacaaIWaaaaaGaayjkaiaawMcaaiabgUca RiabeU7aSnaaBaaaleaacaaIXaaabeaakmaabmaabaqbaeqabmqaaa qaaiaaigdaaeaacaaIXaaabaGaaGymaaaaaiaawIcacaGLPaaaaaa@49B8@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4zaiaaigdacaGG6aGaaGPaVpaaFiaabaGaamiEamaaBaaaleaa caaIXaaabeaaaOGaay51GaGaeyypa0ZaaeWaaeaafaqabeWabaaaba GaaGymaaqaaiaaikdaaeaacaaIWaaaaaGaayjkaiaawMcaaiabgUca RiabeU7aSnaaBaaaleaacaaIXaaabeaakmaabmaabaqbaeqabmqaaa qaaiaaigdaaeaacaaIXaaabaGaaGymaaaaaiaawIcacaGLPaaaaaa@49B7@

g 2 :  x 2 → =( 3 0 2 )+ λ 2 ( 2 0 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4zamaaBaaaleaacaaIYaaabeaakiaacQdacaaMc8+aa8Haaeaa caWG4bWaaSbaaSqaaiaaikdaaeqaaaGccaGLxdcacqGH9aqpdaqada qaauaabeqadeaaaeaacaaIZaaabaGaaGimaaqaaiaaikdaaaaacaGL OaGaayzkaaGaey4kaSIaeq4UdW2aaSbaaSqaaiaaikdaaeqaaOWaae WaaeaafaqabeWabaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaaaaGa ayjkaiaawMcaaaaa@49F3@




Wie groß ist der Abstand der beiden Geraden ? .
Prüfen auf Schnittpunkte und Winkel: .
.
λ1 λ1 c
1 - 2 2 tauschen mit II
1 0 - 2 tauschen mit I
1 - 1 2
1 0 - 2
1 - 2 2 - I
1 - 1 2 - I
1 0 - 2
0 - 2 4 ↯
0 - 1 4 ↯

.
.
Das GLS ist nicht lösbar, es gibt keinen Schnittpunkt. .

Prüfen auf Kollinearität: .
. a 1 → × a 2 → =| x y z 2 1 1 2 0 1 |=( 1−0 2−1 0−2 )=( 1 1 −2 )≠ 0 → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaFiaabaacbi aeaaaaaaaaa8qacaWFHbWdamaaBaaaleaapeGaaGymaaWdaeqaaaGc caGLxdcapeGaey41aq7damaaFiaabaWdbiaa=fgapaWaaSbaaSqaa8 qacaaIYaaapaqabaaakiaawEnia8qacqGH9aqpdaabdaWdaeaafaqa beWadaaabaWdbiaa=Hhaa8aabaWdbiaa=Lhaa8aabaWdbiaa=Phaa8 aabaGaaGOmaaqaa8qacaaIXaaapaqaa8qacaaIXaaapaqaaiaaikda aeaacaaIWaaabaWdbiaaigdaaaaacaGLhWUaayjcSdGaeyypa0Zaae Waa8aabaqbaeqabmqaaaqaa8qacaaIXaGaeyOeI0IaaGimaaWdaeaa peGaaGOmaiabgkHiTiaaigdaa8aabaWdbiaaicdacqGHsislcaaIYa aaaaGaayjkaiaawMcaaiabg2da9maabmaapaqaauaabeqadeaaaeaa peGaaGymaaWdaeaapeGaaGymaaWdaeaapeGaeyOeI0IaaGOmaaaaai aawIcacaGLPaaacqGHGjsUdaWhcaqaaiaaicdaaiaawEniaaaa@6090@ 

| a 1 → × a 2 → |= 1 2 + 1 2 + (−2) 2 = 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaacYhadaWhca qaaGqacabaaaaaaaaapeGaa8xya8aadaWgaaWcbaWdbiaaigdaa8aa beaaaOGaay51GaWdbiabgEna0+aadaWhcaqaa8qacaWFHbWdamaaBa aaleaapeGaaGOmaaWdaeqaaaGccaGLxdcacaGG8bWdbiabg2da9maa kaaabaGaaGymamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdada ahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaeyOeI0IaaGOmaiaa cMcadaahaaWcbeqaaiaaikdaaaaabeaakiabg2da9maakaaabaGaaG OnaaWcbeaaaaa@4DDF@

Abstand: d= | [ a 1 → a 2 → ( r 1 → − r 2 → ) ] | | a 1 → × a 2 → | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadsgacqGH9a qpdaWcaaqaamaaemaabaWaamWaaeaadaWhcaqaaGqacabaaaaaaaaa peGaa8xya8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOGaay51GaWaa8 HaaeaapeGaa8xya8aadaWgaaWcbaWdbiaaikdaa8aabeaaaOGaay51 GaWaaeWaaeaadaWhcaqaaiaadkhadaWgaaWcbaWdbiaaigdaa8aabe aaaOGaay51GaGaeyOeI0Yaa8HaaeaacaWGYbWaaSbaaSqaa8qacaaI YaaapaqabaaakiaawEniaaGaayjkaiaawMcaaaGaay5waiaaw2faaa Gaay5bSlaawIa7aaqaamaaemaabaWaa8HaaeaapeGaa8xya8aadaWg aaWcbaWdbiaaigdaa8aabeaaaOGaay51GaGaey41aq7aa8Haaeaape Gaa8xya8aadaWgaaWcbaWdbiaaikdaa8aabeaaaOGaay51GaaacaGL hWUaayjcSdaaaaaa@5B51@

| [ a 1 → a 2 → ( r 1 → − r 2 → ) ] |=‖ 1 −2 2 1 1 1 2 0 1 ‖=| 2−4+0−4−0+2 |=4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaemaabaWaam WaaeaadaWhcaqaaGqacabaaaaaaaaapeGaa8xya8aadaWgaaWcbaWd biaaigdaa8aabeaaaOGaay51GaWaa8HaaeaapeGaa8xya8aadaWgaa WcbaWdbiaaikdaa8aabeaaaOGaay51GaWaaeWaaeaadaWhcaqaaiaa dkhadaWgaaWcbaWdbiaaigdaa8aabeaaaOGaay51GaGaeyOeI0Yaa8 HaaeaacaWGYbWaaSbaaSqaa8qacaaIYaaapaqabaaakiaawEniaaGa ayjkaiaawMcaaaGaay5waiaaw2faaaGaay5bSlaawIa7aiabg2da9m aafmaabaqbaeqabmWaaaqaaiaaigdaaeaacqGHsislcaaIYaaabaGa aGOmaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaikdaaeaaca aIWaaabaGaaGymaaaaaiaawMa7caGLkWoacqGH9aqpdaabdaqaaiaa ikdacqGHsislcaaI0aGaey4kaSIaaGimaiabgkHiTiaaisdacqGHsi slcaaIWaGaey4kaSIaaGOmaaGaay5bSlaawIa7aiabg2da9iaaisda aaa@6733@
.
d = 4 6. .

.

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