Question

2. Using Levi-Civita symbol, prove (A~ ×B~ )·C~ = (B~ ×C~ )·A~ = (C~ ×A~)·B~

Answer 1

$$({\bf A}\times{\bf B})\cdot{\bf C}=\epsilon_{ijk}A_jB_kC_i=\epsilon_{jki}A_jB_kC_i\\ =\epsilon_{jki}B_kC_iA_j=({\bf B}\times{\bf C})\cdot{\bf A}$$

$$({\bf A}\times{\bf B})\cdot{\bf C}=\epsilon_{ijk}A_jB_kC_i=\epsilon_{kij}A_jB_kC_i\\ =\epsilon_{kij}C_iA_jB_k=({\bf C}\times{\bf A})\cdot{\bf B}$$