cross product evaluations

Question

Within vector calculus I am trying to evaluate the following vector equation. \begin{equation} \boldsymbol A \times (\boldsymbol B \times \boldsymbol C),\qquad \boldsymbol C = \boldsymbol B\times \boldsymbol A \end{equation} My attempts are to evaluate the triple vector product as a dot product but no luck. I believe the result should yield, \begin{equation} \boldsymbol B \times \big( (\boldsymbol A \cdot \boldsymbol A) \cdot \boldsymbol B\big) \end{equation}

Answer 1

$$A \times (B\times C)=A\times(B\times(B\times A))$$ $$=A\times(B(A\cdot B)-A(B\cdot B))$$ $$=(A\cdot B)(A\times B)-(B\cdot B)(A\times A)$$ $$=(A\cdot B)(A\times B)$$