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(D) We use the vector triple product formula: $\overrightarrow{u} 	imes (\overrightarrow{v} 	imes \overrightarrow{w}) = (\overrightarrow{u} \cdot \o

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$(\overrightarrow{a} \cdot \overrightarrow{a})(\overrightarrow{b} 	imes \overrightarrow{a})$

Vedclass · Answer

(D) We use the vector triple product formula: $\overrightarrow{u} 	imes (\overrightarrow{v} 	imes \overrightarrow{w}) = (\overrightarrow{u} \cdot \overrightarrow{w})\overrightarrow{v} - (\overrightarrow{u} \cdot \overrightarrow{v})\overrightarrow{w}$.First,evaluate the inner part: $\overrightarrow{a} 	imes (\overrightarrow{a} 	imes \overrightarrow{b}) = (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{a} - (\overrightarrow{a} \cdot \overrightarrow{a})\overrightarrow{b}$.Now,substitute this back into the original expression:$\overrightarrow{a} 	imes [\overrightarrow{a} 	imes (\overrightarrow{a} 	imes \overrightarrow{b})] = \overrightarrow{a} 	imes [(\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{a} - (\overrightarrow{a} \cdot \overrightarrow{a})\overrightarrow{b}]$$= (\overrightarrow{a} \cdot \overrightarrow{b})(\overrightarrow{a} 	imes \overrightarrow{a}) - (\overrightarrow{a} \cdot \overrightarrow{a})(\overrightarrow{a} 	imes \overrightarrow{b})$Since $\overrightarrow{a} 	imes \overrightarrow{a} = 0$,the first term becomes zero:$= 0 - (\overrightarrow{a} \cdot \overrightarrow{a})(\overrightarrow{a} 	imes \overrightarrow{b})$$= (\overrightarrow{a} \cdot \overrightarrow{a})(\overrightarrow{b} 	imes \overrightarrow{a})$.

$\overrightarrow{a} \times [\overrightarrow{a} \times (\overrightarrow{a} \times \overrightarrow{b})]$ is equal to

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