For any four vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}, \vec{d} .(\vec{a} \times(\vec{b} \times(\vec{c} \times \vec{d})))$ is always equal to:
Answer & explanation
Correct answer: option 2
$\vec{d} .(\vec{a} \times(\vec{b} \times(\vec{c} \times \vec{d})))$
$=\vec{d} .(\vec{a} \times((\vec{b} . \vec{d}) \vec{c}-(\vec{b} . \vec{c}) \vec{d}))$
$=\vec{d} .((\vec{b} . \vec{d})(\vec{a} \times \vec{c})-(\vec{b} . \vec{c})(\vec{a} \times \vec{d}))$
$=(\vec{b} . \vec{d})(\vec{d} .(\vec{a} \times \vec{c})-(\vec{c})(\vec{a} \times\vec{d}))$
$=(\vec{b} . \vec{d})[\vec{d} \vec{a} \vec{c}]$
$=(\vec{b} . \vec{d})[\vec{a} \vec{c} \vec{d}]$
Hence (2) is correct answer.