0

0/19

20 Anwendungen

0/19/3

20.3 Vektorielle Darstellung der Ebene

0/19/3/3

20.3.3 Gleichung einer Ebene senkrecht zu einem Vektor (Normalenvektordarstellung)

0/19/3/3/3 .
Beispiel 20 - 214
Umwandlung in eine Normalenvektor-Darstellung .
Gegeben sei eine Ebene .

r (P)=( 3 5 1 )+λ( 5 6 7 )+μ( 1 5 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaabaaaaaaaaape Waa8HaaeaacaWGYbaacaGLxdcacaGGOaGaamiuaiaacMcacqGH9aqp daqadaqaauaabeqadeaaaeaacaaIZaaabaGaaGynaaqaaiaaigdaaa aacaGLOaGaayzkaaGaey4kaSIaeq4UdW2aaeWaaeaafaqabeWabaaa baGaeyOeI0IaaGynaaqaaiabgkHiTiaaiAdaaeaacaaI3aaaaaGaay jkaiaawMcaaiabgUcaRiabeY7aTnaabmaabaqbaeqabmqaaaqaaiab gkHiTiaaigdaaeaacqGHsislcaaI1aaabaGaaGimaaaaaiaawIcaca GLPaaaaaa@503B@  ..

Ein Normalenvektor ist z.B.#
n =                                       #
                                      a ×                                                                                                                                    #
                                                                                                                                    b .

 .

n = a × b =| x y z 5 6 7 1 5 0 |=( 0+35 7+0 256 )=( 35 7 19 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaFiaabaGaam OBaaGaay51GaGaeyypa0Zaa8HaaeaacaWGHbaacaGLxdcacqGHxdaT daWhcaqaaiaadkgaaiaawEniaiabg2da9maaemaabaqbaeqabmWaaa qaaiaadIhaaeaacaWG5baabaGaamOEaaqaaiabgkHiTiaaiwdaaeaa cqGHsislcaaI2aaabaGaaG4naaqaaiabgkHiTiaaigdaaeaacqGHsi slcaaI1aaabaGaaGimaaaaaiaawEa7caGLiWoacqGH9aqpdaqadaqa auaabeqadeaaaeaacaaIWaGaey4kaSIaaG4maiaaiwdaaeaacqGHsi slcaaI3aGaey4kaSIaaGimaaqaaiaaikdacaaI1aGaeyOeI0IaaGOn aaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqadeaaaeaaca aIZaGaaGynaaqaaiabgkHiTiaaiEdaaeaacaaIXaGaaGyoaaaaaiaa wIcacaGLPaaaaaa@63AF@ 

n (r -r1) = 0 .
.
35-7 19 rx - 3 ry - 5 rz - 1 = 0.
.
Umwandlung in Achsenabschnittsform: .
35(rx - 3) - 7(ry - 5) + 19(rz - 1) = 0 .
35rx - 7ry + 19rz = 105 - 35 + 19 = 99 .

.