Statement-1: Let $V_1$ be the volume of a parallelopiped ABCDEF having $\vec a,\vec b,\vec c$ as three coterminous edges and $V_2$ be the volume of the parallelopiped PQRSTU having three coterminous edges as vectors whose magnitudes are equal to the areas of three adjacent faces of the parallelopiped ABCDEF. Then, $V_2=2V_1^2$.
Statement-2: For any three vectors $\vec α,\vec β,\vec γ$
$\begin{bmatrix}\vec α×\vec β&\vec β×\vec γ&\vec γ×α\end{bmatrix}=[\vec α\,\,\vec β\,\,\vec γ]^2$
Answer & explanation
Correct answer: option 4
Clearly, statement-2 is true.
Three coterminous edges of parallelopiped PQRSTU are $\vec a×\vec b, \vec b×\vec c$ and $\vec c×\vec a$.
$∴V_2=\left|\begin{bmatrix}\vec a×\vec b&\vec b×\vec c&\vec c×\vec a\end{bmatrix}\right|$
$⇒V_2=\left|[\vec a\,\,\vec b\,\,\vec c]^2\right|$ [Using statement-2]
$⇒V_2=\left|[\vec a\,\,\vec b\,\,\vec c]\right|^2⇒V_2=V_1^2$