משתמש:Eman/ארגז חול

$(A\times B)_{1}=\sum _{j,k=1}^{3}\epsilon _{1jk}A_{j}B_{k}$

$=\epsilon _{111}A_{1}B_{1}+\epsilon _{112}A_{1}B_{2}+\epsilon _{113}A_{1}B_{3}+\epsilon _{121}A_{2}B_{1}+\epsilon _{122}A_{2}B_{2}+\epsilon _{123}A_{2}B_{3}+\epsilon _{131}A_{3}B_{1}+\epsilon _{132}A_{3}B_{2}+\epsilon _{133}A_{3}B_{3}$

$=0\cdot A_{1}B_{1}+0\cdot A_{1}B_{2}+0\cdot A_{1}B_{3}+0\cdot A_{2}B_{1}+0\cdot A_{2}B_{2}+1\cdot A_{2}B_{3}+0\cdot A_{3}B_{1}+(-1)\cdot A_{3}B_{2}+0\cdot A_{3}B_{3}$

$=A_{2}B_{3}-A_{3}B_{2}$