2.1 Allgemeine Beziehungen

Im folgenden einige grundlegende Gleichungen der Vektorrechnung und Beziehungen zwischen den Vektorprodukten^2.1

$\displaystyle \vec{A}\,$ ^. $\displaystyle \vec{B}\,$ `x` $\displaystyle \vec{C}\,$ = $\displaystyle \vec{A}\,$ `x` $\displaystyle \vec{B}\,$ ^. $\displaystyle \vec{C}\,$ $\displaystyle \equiv$ ( $\displaystyle \vec{A}\,$ , $\displaystyle \vec{B}\,$ , $\displaystyle \vec{C}\,$ )=( $\displaystyle \vec{B}\,$ , $\displaystyle \vec{C}\,$ , $\displaystyle \vec{A}\,$ )=( $\displaystyle \vec{C}\,$ , $\displaystyle \vec{A}\,$ , $\displaystyle \vec{B}\,$ )=
-( $\displaystyle \vec{A}\,$ , $\displaystyle \vec{C}\,$ , $\displaystyle \vec{B}\,$ )=-( $\displaystyle \vec{C}\,$ , $\displaystyle \vec{B}\,$ , $\displaystyle \vec{A}\,$ )=-( $\displaystyle \vec{B}\,$ , $\displaystyle \vec{A}\,$ , $\displaystyle \vec{C}\,$ )

$\displaystyle \vec{A}\,$ x ( $\displaystyle \vec{B}\,$ x $\displaystyle \vec{C}\,$ ) = ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \vec{C}\,$ ) $\displaystyle \vec{B}\,$ - ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \vec{B}\,$ ) $\displaystyle \vec{C}\,$ (2.2)

$\displaystyle \vec{A}\,$ x ( $\displaystyle \vec{B}\,$ x $\displaystyle \vec{C}\,$ ) + $\displaystyle \vec{B}\,$ x ( $\displaystyle \vec{C}\,$ x $\displaystyle \vec{A}\,$ ) + $\displaystyle \vec{C}\,$ x ( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ) = 0 (2.3)

( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ )^.( $\displaystyle \vec{C}\,$ x $\displaystyle \vec{D}\,$ ) = $\displaystyle \vec{A}\,$ ^.( $\displaystyle \vec{B}\,$ x ( $\displaystyle \vec{C}\,$ x $\displaystyle \vec{D}\,$ )) = ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \vec{B}\,$ )( $\displaystyle \vec{B}\,$ ^. $\displaystyle \vec{D}\,$ ) - ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \vec{D}\,$ )( $\displaystyle \vec{B}\,$ ^. $\displaystyle \vec{D}\,$ ) (2.4)

( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ) x ( $\displaystyle \vec{C}\,$ x $\displaystyle \vec{D}\,$ ) = ( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ^. $\displaystyle \vec{D}\,$ ) $\displaystyle \vec{C}\,$ - ( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ^. $\displaystyle \vec{C}\,$ ) $\displaystyle \vec{D}\,$ (2.5)

( $\displaystyle \vec{a}\,$ `x` $\displaystyle \vec{b}\,$ )_i	=	$\displaystyle \varepsilon_{ijk}^{}$ a_jb_k	(2.6)
$\displaystyle \varepsilon_{ijk}^{}$	=	$\displaystyle \varepsilon_{kji}^{}$ tex2html_image_mark>#tex2html_wrap_indisplay3321# zyklisch	(2.7)
$\displaystyle \varepsilon_{ijk}^{}$	=	- $\displaystyle \varepsilon_{jik}^{}$ tex2html_image_mark>#tex2html_wrap_indisplay3327# antisymmetrisch	(2.8)
$\displaystyle \varepsilon_{kij}^{}$ $\displaystyle \varepsilon_{klm}^{}$	=	$\displaystyle \delta_{il}^{}$ $\displaystyle \delta_{jm}^{}$ - $\displaystyle \delta_{im}^{}$ $\displaystyle \delta_{jl}^{}$	(2.9)