a¯×{a¯×(a¯×b¯)}=

a¯×(a¯×(a¯×b¯))=a¯×[(a¯⋅b¯)a¯−(a¯⋅a¯)b¯]=(a¯⋅b¯)(a¯×a¯)−(a¯⋅a¯)(a¯×b¯)=(a¯⋅a¯)(b¯×a¯)

a¯×{a¯×(a¯×b¯)}=

a¯×(a¯×(a¯×b¯))=a¯×[(a¯⋅b¯)a¯−(a¯⋅a¯)b¯]=(a¯⋅b¯)(a¯×a¯)−(a¯⋅a¯)(a¯×b¯)=(a¯⋅a¯)(b¯×a¯)

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$\bar{a} \times {\bar{a} \times (\bar{a} \times \bar{b})} =$

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$(\bar{a} . \bar{a}) (\bar{b} \times \bar{a})$

$(\bar{b} . \bar{b}) (\bar{a} \times \bar{b})$

$(\bar{a} \cdot \bar{b}) (\bar{a} \times \bar{b})$

$(\bar{a} . \bar{a}) (\bar{a} \times \bar{b})$

answer is C.

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Detailed Solution

$\begin{array}{l} \bar{a} \times (\bar{a} \times (\bar{a} \times \bar{b})) \\ = \bar{a} \times [(\bar{a} \cdot \bar{b}) \bar{a} - (\bar{a} \cdot \bar{a}) \bar{b}] \\ = (\bar{a} \cdot \bar{b}) (\bar{a} \times \bar{a}) - (\bar{a} \cdot \bar{a}) (\bar{a} \times \bar{b}) \\ = (\bar{a} \cdot \bar{a}) (\bar{b} \times \bar{a}) \end{array}$