Let $\vec{a}, \vec{b}, \vec{c}$ be any three vectors. Then $[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}]$ is also equal to:
Answer & explanation
Correct answer: option 1
$[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}]$
$=(\vec{a} \times \vec{b}) . ((\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a}))$
$=(\vec{a} \times \vec{b}) . ([\vec{b} \vec{c} \vec{a}] \vec{c}-[\vec{b} \vec{c} \vec{c}] \vec{a})$
$=[\vec{a} \vec{b} \vec{c}]^2$
Hence (1) is correct answer.