Eigenschaften des Vektorprodukts



Für x,y 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyicI4SaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3C72@ ist das Vektorprodukt von x und y gegeben durch
 

x×y=( x 2 y 3 x 3 y 2 x 1 y 3 + x 3 y 1 x 1 y 2 x 2 y 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@577C@  .

Die konkrete Berechnung von Zahlenbeispielen kann man durch ein Merkschema übersichtlich organisieren. Schreibt man statt des Ausdrucks adcb MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadsgacqGHsislcaWGJbGaamOyaaaa@3A74@ das Symbol 
a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36CF@ × b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36D0@
c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36D1@ d MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaaaa@36D2@
 , also:

Produkt in der Hauptdiagonale minus Produkt in der Nebendiagonale,

so errechnen sich die drei Koordinaten des Vektors x×y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadMhaaaa@39FB@ der Reihe nach zu:
 
x 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaaaa@37CD@   y 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaaaaa@37CE@
x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaaaaa@37CE@ × y 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@37CF@
x 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIZaaabeaaaaa@37CF@ y 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIZaaabeaaaaa@37D0@
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0caaa@36D6@
x 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaaaa@37CD@ × y 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaaaaa@37CE@
x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaaaaa@37CE@ y 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@37CF@
x 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIZaaabeaaaaa@37CF@ y 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIZaaabeaaaaa@37D0@
x 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaaaa@37CD@ × y 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaaaaa@37CE@
x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaaaaa@37CE@ y 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@37CF@
x 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIZaaabeaaaaa@37CF@   y 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIZaaabeaaaaa@37D0@
 .

 
Beispiel:  Um etwa den Vektor ( 3 1 2 )×( 1 4 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaG4maaqaaiabgkHiTiaaigdaaeaacaaIYaaaaaGaayjkaiaawMcaaiabgEna0oaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaI0aaabaGaaGimaaaaaiaawIcacaGLPaaaaaa@4084@ zu berechnen, ermittelt man zunächst
 
1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaGymaaaa@3791@ × 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@36A7@
2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@36A5@ 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36A3@
=8 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaeyOeI0IaaGioaaaa@389E@
3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@36A6@ × 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@36A4@
2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@36A5@ 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36A3@
=2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaeyOeI0IaaGOmaaaa@3898@
3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@36A6@ × 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@36A4@
1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaGymaaaa@3791@ 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@36A7@
=13 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaGymaiaaiodaaaa@3867@

und erhält so: ( 3 1 2 )×( 1 4 0 )=( 8 2 13 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaG4maaqaaiabgkHiTiaaigdaaeaacaaIYaaaaaGaayjkaiaawMcaaiabgEna0oaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaI0aaabaGaaGimaaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqadeaaaeaacqGHsislcaaI4aaabaGaaGOmaaqaaiaaigdacaaIZaaaaaGaayjkaiaawMcaaaaa@4705@ .  

Für theoretische Überlegungen interessanter ist die Möglichkeit, x×y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadMhaaaa@39FB@ über eine Matrixanwendung zu berechnen. Ordnen wir nämlich jedem Vektor x 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maaaaaaa@3AC4@ die Matrix
 

( a ij ) x =( 0 x 3 x 2 x 3 0 x 1 x 2 x 1 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaamaaBaaaleaacaWG4baabeaakiabg2da9maabmaabaqbaeqabmWaaaqaaiaaicdaaeaacqGHsislcaWG4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaaakeaacaaIWaaabaGaeyOeI0IaamiEamaaBaaaleaacaaIXaaabeaaaOqaaiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaGimaaaaaiaawIcacaGLPaaaaaa@4ED3@

zu, so gilt offensichtlich:  x×y= ( a ij ) x y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadMhacqGH9aqpdaqadaqaaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaadIhaaeqaaOGaamyEaaaa@41B4@ .

 

Bemerkung:  Für x,y,z 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaaiilaiaadQhacqGHiiIZcqWIDesOdaahaaWcbeqaaiaaiodaaaaaaa@3E21@ , α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHekaaa@3A7C@ gilt:
  1. x×y=(y×x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadMhacqGH9aqpcqGHsislcaGGOaGaamyEaiabgEna0kaadIhacaGGPaaaaa@4159@
     
  2. x×x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadIhacqGH9aqpcaaIWaaaaa@3BBA@
     
  3. x×(y+z)=x×y+x×z MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaacIcacaWG5bGaey4kaSIaamOEaiaacMcacqGH9aqpcaWG4bGaey41aqRaamyEaiabgUcaRiaadIhacqGHxdaTcaWG6baaaa@4742@
     
  4. x×(αy)=α(x×y) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaacIcacqaHXoqycaWG5bGaaiykaiabg2da9iabeg7aHjaacIcacaWG4bGaey41aqRaamyEaiaacMcaaaa@4503@
     

Beweis:

Zu 1.:  x×y=( x 2 y 3 x 3 y 2 x 1 y 3 + x 3 y 1 x 1 y 2 x 2 y 1 )=( y 2 x 3 y 3 x 2 y 1 x 3 + y 3 x 1 y 1 x 2 y 2 x 1 )=(y×x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7D48@ .

Zu 2.:  Folgt direkt aus x×x=(x×x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadIhacqGH9aqpcqGHsislcaGGOaGaamiEaiabgEna0kaadIhacaGGPaaaaa@4157@ .

Zu 3.:  Matrizen operieren linear. Man also: x×(y+z)= ( a ij ) x (y+z)= ( a ij ) x y+ ( a ij ) x z=x×y+x×z MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaacIcacaWG5bGaey4kaSIaamOEaiaacMcacqGH9aqpdaqadaqaaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaadIhaaeqaaOGaaiikaiaadMhacqGHRaWkcaWG6bGaaiykaiabg2da9maabmaabaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaadaWgaaWcbaGaamiEaaqabaGccaWG5bGaey4kaSYaaeWaaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaamaaBaaaleaacaWG4baabeaakiaadQhacqGH9aqpcaWG4bGaey41aqRaamyEaiabgUcaRiaadIhacqGHxdaTcaWG6baaaa@6184@ .

Zu 4.:  x×(αy)= ( a ij ) x (αy)=α ( a ij ) x y=α(x×y) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaacIcacqaHXoqycaWG5bGaaiykaiabg2da9maabmaabaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaadaWgaaWcbaGaamiEaaqabaGccaGGOaGaeqySdeMaamyEaiaacMcacqGH9aqpcqaHXoqydaqadaqaaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaadIhaaeqaaOGaamyEaiabg2da9iabeg7aHjaacIcacaWG4bGaey41aqRaamyEaiaacMcaaaa@590C@ .
 

Beachte:

Bemerkung:  Für v,w,x,y,z 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaaiilaiaadIhacaGGSaGaamyEaiaacYcacaWG6bGaeyicI4SaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@4178@ gilt:
  1. (x×y)×z=(x·z)y(y·z)x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHxdaTcaWG5bGaaiykaiabgEna0kaadQhacqGH9aqpcaGGOaGaamiEaiabl+y6NjaadQhacaGGPaGaamyEaiabgkHiTiaacIcacaWG5bGaeS4JPFMaamOEaiaacMcacaWG4baaaa@4DE3@                                          ("Grassmannscher Entwicklungssatz")
     
  2. (y×z)×x+(z×x)×y+(x×y)×z=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadMhacqGHxdaTcaWG6bGaaiykaiabgEna0kaadIhacqGHRaWkcaGGOaGaamOEaiabgEna0kaadIhacaGGPaGaey41aqRaamyEaiabgUcaRiaacIcacaWG4bGaey41aqRaamyEaiaacMcacqGHxdaTcaWG6bGaeyypa0JaaGimaaaa@52F0@                           ("Jacobi-Identität")
     
  3. (x×y)·z=(y×z)·x=(z×x)·y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHxdaTcaWG5bGaaiykaiabl+y6NjaadQhacqGH9aqpcaGGOaGaamyEaiabgEna0kaadQhacaGGPaGaeS4JPFMaamiEaiabg2da9iaacIcacaWG6bGaey41aqRaamiEaiaacMcacqWIpM+zcaWG5baaaa@5283@
     
  4. (x×y)·(v×w)=(x·v)(y·w)(x·w)(v·y) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHxdaTcaWG5bGaaiykaiabl+y6NjaacIcacaWG2bGaey41aqRaam4DaiaacMcacqGH9aqpcaGGOaGaamiEaiabl+y6NjaadAhacaGGPaGaaiikaiaadMhacqWIpM+zcaWG3bGaaiykaiabgkHiTiaacIcacaWG4bGaeS4JPFMaam4DaiaacMcacaGGOaGaamODaiabl+y6NjaadMhacaGGPaaaaa@5C26@               ("Lagrangesche Identität")
     
  5. (x×y) 2 = x 2 y 2 (x·y) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHxdaTcaWG5bGaaiykamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaiikaiaadIhacqWIpM+zcaWG5bGaaiykamaaCaaaleqabaGaaGOmaaaaaaa@48C8@

Beweis:

Zu 1.:  Wir benutzen die Zerlegung
 

( a ij ) x×y =( 0 x 1 y 2 + x 2 y 1 x 1 y 3 + x 3 y 1 x 1 y 2 x 2 y 1 0 x 2 y 3 + x 3 y 2 x 1 y 3 x 3 y 1 x 2 y 3 x 3 y 2 0 )=( 0 x 2 y 1 x 3 y 1 x 1 y 2 0 x 3 y 2 x 1 y 3 x 2 y 3 0 )( 0 x 1 y 2 x 1 y 3 x 2 y 1 0 x 2 y 3 x 3 y 1 x 3 y 2 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B232@

und berechnen damit:
 
(x×y)×z = ( a ij ) x×y z =( 0 x 2 y 1 x 3 y 1 x 1 y 2 0 x 3 y 2 x 1 y 3 x 2 y 3 0 )z( 0 x 1 y 2 x 1 y 3 x 2 y 1 0 x 2 y 3 x 3 y 1 x 3 y 2 0 )z =( ( x 2 z 2 + x 3 z 3 ) y 1 ( x 1 z 1 + x 3 z 3 ) y 2 ( x 1 z 1 + x 2 z 2 ) y 3 )( ( y 2 z 2 + y 3 z 3 ) x 1 ( y 1 z 1 + y 3 z 3 ) x 2 ( y 1 z 1 + y 2 z 2 ) x 3 ) =(x·z)y( x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 )((y·z)x( x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 )) =(x·z)y(y·z)x. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabuGaaaaabaGaaiikaiaadIhacqGHxdaTcaWG5bGaaiykaiabgEna0kaadQhaaeaacqGH9aqpdaqadaqaaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaadIhacqGHxdaTcaWG5baabeaakiaadQhaaeaaaeaacqGH9aqpdaqadaqaauaabeqadmaaaeaacaaIWaaabaGaamiEamaaBaaaleaacaaIYaaabeaakiaadMhadaWgaaWcbaGaaGymaaqabaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG5bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaGimaaqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccaWG5bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaadMhadaWgaaWcbaGaaG4maaqabaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamyEamaaBaaaleaacaaIZaaabeaaaOqaaiaaicdaaaaacaGLOaGaayzkaaGaamOEaiabgkHiTmaabmaabaqbaeqabmWaaaqaaiaaicdaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamyEamaaBaaaleaacaaIYaaabeaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG5bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaakiaadMhadaWgaaWcbaGaaGymaaqabaaakeaacaaIWaaabaGaamiEamaaBaaaleaacaaIYaaabeaakiaadMhadaWgaaWcbaGaaG4maaqabaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccaWG5bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaGimaaaaaiaawIcacaGLPaaacaWG6baabaaabaGaeyypa0ZaaeWaaeaafaqabeWabaaabaGaaiikaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaWG6bWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamiEamaaBaaaleaacaaIZaaabeaakiaadQhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamOEamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaWG6bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaadMhadaWgaaWcbaGaaGOmaaqabaaakeaacaGGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaadQhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamOEamaaBaaaleaacaaIYaaabeaakiaacMcacaWG5bWaaSbaaSqaaiaaiodaaeqaaaaaaOGaayjkaiaawMcaaiabgkHiTmaabmaabaqbaeqabmqaaaqaaiaacIcacaWG5bWaaSbaaSqaaiaaikdaaeqaaOGaamOEamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadMhadaWgaaWcbaGaaG4maaqabaGccaWG6bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaadIhadaWgaaWcbaGaaGymaaqabaaakeaacaGGOaGaamyEamaaBaaaleaacaaIXaaabeaakiaadQhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWG5bWaaSbaaSqaaiaaiodaaeqaaOGaamOEamaaBaaaleaacaaIZaaabeaakiaacMcacaWG4bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaiikaiaadMhadaWgaaWcbaGaaGymaaqabaGccaWG6bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyEamaaBaaaleaacaaIYaaabeaakiaadQhadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaamiEamaaBaaaleaacaaIZaaabeaaaaaakiaawIcacaGLPaaaaeaaaeaacqGH9aqpcaGGOaGaamiEaiabl+y6NjaadQhacaGGPaGaamyEaiabgkHiTmaabmaabaqbaeqabmqaaaqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaamOEamaaBaaaleaacaaIXaaabeaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaWG5bWaaSbaaSqaaiaaikdaaeqaaOGaamOEamaaBaaaleaacaaIYaaabeaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccaWG5bWaaSbaaSqaaiaaiodaaeqaaOGaamOEamaaBaaaleaacaaIZaaabeaaaaaakiaawIcacaGLPaaacqGHsislcaGGOaGaaiikaiaadMhacqWIpM+zcaWG6bGaaiykaiaadIhacqGHsisldaqadaqaauaabeqadeaaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamyEamaaBaaaleaacaaIXaaabeaakiaadQhadaWgaaWcbaGaaGymaaqabaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamyEamaaBaaaleaacaaIYaaabeaakiaadQhadaWgaaWcbaGaaGOmaaqabaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaamyEamaaBaaaleaacaaIZaaabeaakiaadQhadaWgaaWcbaGaaG4maaqabaaaaaGccaGLOaGaayzkaaGaaiykaaqaaaqaaiabg2da9iaacIcacaWG4bGaeS4JPFMaamOEaiaacMcacaWG5bGaeyOeI0IaaiikaiaadMhacqWIpM+zcaWG6bGaaiykaiaadIhaaaaaaa@182F@

Zu 2.:  Wir benutzen den Grassmannschen Satz und erhalten für die drei Summanden:

(y×z)×x=(y·x)z(z·x)y (z×x)×y=(z·y)x(x·y)z (x×y)×z=(x·z)y(y·z)x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7DE6@
 
Da das Skalarprodukt kommutativ ist, heben sich die sechs rechts stehenden Summanden auf. Das Ergebnis ist also der Nullvektor.

Zu 3.:  Es reicht, die erste Gleichheit zu zeigen (Siehe auch den nachfolgenden Kommentar). Das Überfahren der zweiten und dritten Zeile mit dem Mauszeiger unterstützt die Überprüfung der Identität.
 

(x×y)·z MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHxdaTcaWG5bGaaiykaiabl+y6NjaadQhaaaa@3EC3@ =( x 2 y 3 x 3 y 2 x 1 y 3 + x 3 y 1 x 1 y 2 x 2 y 1 )z MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5469@
  = MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36EF@ x 2 y 3 z 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaakiaadMhadaWgaaWcbaGaaG4maaqabaGccaWG6bWaaSbaaSqaaiaaigdaaeqaaaaa@3BAF@ x 3 y 2 z 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiEamaaBaaaleaacaaIZaaabeaakiaadMhadaWgaaWcbaGaaGOmaaqabaGccaWG6bWaaSbaaSqaaiaaigdaaeqaaaaa@3C9C@ x 1 y 3 z 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiEamaaBaaaleaacaaIXaaabeaakiaadMhadaWgaaWcbaGaaG4maaqabaGccaWG6bWaaSbaaSqaaiaaikdaaeqaaaaa@3C9C@ + x 3 y 1 z 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaamiEamaaBaaaleaacaaIZaaabeaakiaadMhadaWgaaWcbaGaaGymaaqabaGccaWG6bWaaSbaaSqaaiaaikdaaeqaaaaa@3C91@ + x 1 y 2 z 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaamiEamaaBaaaleaacaaIXaaabeaakiaadMhadaWgaaWcbaGaaGOmaaqabaGccaWG6bWaaSbaaSqaaiaaiodaaeqaaaaa@3C91@ x 2 y 1 z 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiEamaaBaaaleaacaaIYaaabeaakiaadMhadaWgaaWcbaGaaGymaaqabaGccaWG6bWaaSbaaSqaaiaaiodaaeqaaaaa@3C9C@
  = MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36EF@ y 2 z 3 x 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaakiaadQhadaWgaaWcbaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaaaa@3BAF@ y 3 z 2 x 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamyEamaaBaaaleaacaaIZaaabeaakiaadQhadaWgaaWcbaGaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaaaa@3C9C@ y 1 z 3 x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamyEamaaBaaaleaacaaIXaaabeaakiaadQhadaWgaaWcbaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaaaa@3C9C@ + y 3 z 1 x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaamyEamaaBaaaleaacaaIZaaabeaakiaadQhadaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaaaa@3C91@ + y 1 z 2 x 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaamyEamaaBaaaleaacaaIXaaabeaakiaadQhadaWgaaWcbaGaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@3C91@ y 2 z 1 x 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamyEamaaBaaaleaacaaIYaaabeaakiaadQhadaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@3C9C@
  =( y 2 z 3 y 3 z 2 y 1 z 3 + y 3 z 1 y 1 z 2 y 2 z 1 )x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5473@

 

=(y×z)·x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaiikaiaadMhacqGHxdaTcaWG6bGaaiykaiabl+y6NjaadIhaaaa@3FC9@ .
 

Zu 4.:  Die Lagrangesche Identität erhalten wir aus 1. und 2.:

(x×y)·(v×w) =(y×(v×w))·x =((v×w)×x)·y =((v·x)w(w·x)v)·y =(v·x)(w·y)(w·x)(v·y). MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8DB1@

Die angegebene Gleichheit folgt nun mit der Kommutativität des Skalarprodukts.

Zu 5.:  Dies ist ein Spezialfall von 3. Man setzt dort v=x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iaadIhaaaa@38E7@ und w=y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Daiabg2da9iaadMhaaaa@38E9@ .
  

Beachte:

Bemerkung:  Für x,y 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyicI4SaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3C72@ gilt:
  1. x×yxx×yy MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadMhacqGHLkIxcaWG4bGaaGzbVlabgEIizlaaywW7caWG4bGaey41aqRaamyEaiabgwQiEjaadMhaaaa@4834@
     
  2. |x×y|=|x||y|sin((x,y)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHxdaTcaWG5bGaaiiFaiabg2da9iaacYhacaWG4bGaaiiFaiaacYhacaWG5bGaaiiFaiGacohacaGGPbGaaiOBaiaacIcacqWIIiYucaGGOaGaamiEaiaacYcacaWG5bGaaiykaiaacMcaaaa@4C59@
     
  3. x×y=0x,y   linear abhängig MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEna0kaadMhacqGH9aqpcaaIWaGaaGzbVlabgsDiBlaaywW7caWG4bGaaiilaiaadMhacaaMe8UaaeiBaiaabMgacaqGUbGaaeyzaiaabggacaqGYbGaaeiiaiaabggacaqGIbGaaeiAaiaabsoacaqGUbGaae4zaiaabMgacaqGNbaaaa@5367@

Beweis:

Zu 1.:  Es reicht, nur die erste Orthogonalität nachzuweisen. Nach der gerade notierten Eigenschaft des Spatprodukts hat man
 

(x×y)·x=(x×x)·y=0·y=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHxdaTcaWG5bGaaiykaiabl+y6NjaadIhacqGH9aqpcaGGOaGaamiEaiabgEna0kaadIhacaGGPaGaeS4JPFMaamyEaiabg2da9iaaicdacqWIpM+zcaWG5bGaeyypa0JaaGimaaaa@4F8D@ .

Zu 2.:  Nach 4. in der Bemerkung zuvor hat man:

|x×y | 2 =|x | 2 |y | 2 (x·y) 2 =|x | 2 |y | 2 |x | 2 |y | 2 cos 2 ((x,y)) =|x | 2 |y | 2 (1 cos 2 ((x,y))) =|x | 2 |y | 2 sin 2 ((x,y)). MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@93AA@

Aus der Gleichheit der Quadrate folgt direkt die Behauptung (Beachte: sin((x,y))0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiikaiablkIitjaacIcacaWG4bGaaiilaiaadMhacaGGPaGaaiykaiabgwMiZkaaicdaaaa@41C6@ ).

Zu 3.:  Wir argumentieren ähnlich wie gerade:
 

x×y=0    |x×y | 2 =0 |x | 2 |y | 2 = (x·y) 2 |x||y|=|x·y| x,y   linear abhängig. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@85B9@

Die letzte Äquivalenz ergibt sich dabei aus einem Zusatz zur Cauchy-Schwarzschen Ungleichung aus 9.13.
 

Beachte: