Practicing Success
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 $\vec α,\vec β,\vec γ$ $\begin{bmatrix}\vec α×\vec β&\vec β×\vec γ&\vec γ×α\end{bmatrix}=[\vec α\,\,\vec β\,\,\vec γ]^2$ |
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1. Statement-1 is True, Statement-2 is False. Statement-1 is False, Statement-2 is True. |
Statement-1 is False, Statement-2 is True. |
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] $⇒V_2=\left|[\vec a\,\,\vec b\,\,\vec c]\right|^2⇒V_2=V_1^2$ |