In the following questions, a statement of assertion (Statement-1) is followed by a statement of reason (Statement-2). Mark the correct choice as :
Statement-1 : Let $\vec{a}, \vec{b}, \vec{c}$ be three coterminous edges of a parallelepiped of volume $V$ . Let $V 1$ be the volume of the parallelepiped whose three coterminous edges are the diagonals of three adjacent faces of the given parallelopiped. Then, $V_{1} = 2 V$ .
Statement-2 : For any three vectors $\vec{a}, \vec{b}, \vec{c}$ $[\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = 2 [\vec{a} \vec{b} \vec{c}]$

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If both Statement-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1.

If both Statement-1 and Statement-2 are true but Statement-2 is not the correct explanation of Statement-1.

If Statement-1 is true but Statement-2 is false.

If Statement-1 is true but Statement-2 is true.

answer is A.

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Detailed Solution

Let $\vec{α} = \vec{a} + \vec{b}, \vec{β} = \vec{b} + \vec{c}$ and $\vec{γ} = \vec{c} + \vec{a}$ . Then, $\vec{α} = \vec{a} + \vec{b} + 0, \vec{β} = 0 \vec{a} + \vec{b} + \vec{c}$ and

$[\vec{α} \vec{β} \vec{γ}] = |\begin{array}{lll} 1 1 0 \\ 0 1 1 \\ 1 0 1 \end{array}| [\vec{a} \vec{b} \vec{c}] \Rightarrow [\vec{α} \vec{β} \vec{γ}] = 2 [\vec{a} \vec{b} \vec{c}]$

So, statement- $2$ is true
We have, $V = [\vec{a} \vec{b} \vec{c}]$ and $V_{1} = | [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] |$ Using statement- $2$ , we have
$V_{1} = 2 [\vec{a} \vec{b} \vec{c}] = 2 V .$
So, statement-1 is true and statement- $2$ is a correct explanation for statement- $1$ .