HW1 Answer

1.1~1.7 %

\begin{aligned} LHS & = b \cdot (- I \times a) = - b \times a = a \times b \\ = a \times I \cdot b = RHS \end{aligned}

\begin{aligned} LHS & = (a \times I) \cdot (I \times a) \\ = a \times [I \cdot (I \times a)] \\ = a \times I \times a \\ = (ϵ_{i j k} a_{i} δ_{j n} {\hat{e}}_{k} {\hat{e}}_{n}) \times a_{m} {\hat{e}}_{m} = ϵ_{i j k} ϵ_{n m l} δ_{j n} a_{i} a_{m} {\hat{e}}_{k} \hat{e_{l}} = - ϵ_{n i k} ϵ_{n m l} a_{i} a_{m} {\hat{e}}_{k} {\hat{e}}_{l} \\ = - [a_{m} a_{m} {\hat{e}}_{k} {\hat{a}}_{l} - a_{l} a_{k} {\hat{e}}_{k} {\hat{a}}_{l}] \\ = RHS \end{aligned}