Volume of parallelopiped formed by vectors $\to a \times \to b, \to b \times \to c$ and $\to c \times \to a$ is $36$ cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.

$\begin{matrix} C o l u m n 1 & C o l u m n 2 a. Volume of parallelopiped formed by vectors & p. 0 cubic units \to a, \to b and \to c is b. Volume of tetrahedron formed by vectors & q. 12 cubic units \to a, \to b and \to c is c. Volume of parallelopiped formed by vectors & r. 6 cubic units \to a + \to b, \to b + \to c and \to c + \to a is d. Volume of parallelopiped formed by vectors & s. 1 cubic unit \to a - \to b, \to b - \to c and \to c - \to a is \end{matrix}$

a - r, b - s, c - q, d - p

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a - p, b - s, c - q, d - r

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a - r, b - q, c - s, d - p

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a - s, b - r, c - q, d - p

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Solution

The correct option is A $a - r, b - s, c - q, d - p$
$[\to a \times \to b \to b \times \to c \to c \times \to a] = 36$
$[\to a \to b \to c] = 6$
$\Rightarrow$ Volume of tetrahedron formed by vectors $\to a, \to b$ and $\to c$ is $\frac{1}{6} [\to a \to b \to c] = 1$
$[\to a + \to b \to b + \to c \to c + \to a] = 2 [\to a \to b \to c] = 12$
$\to a - \to b, \to b - \to c, \to c - \to a$ are coplanar
$[\to a - \to b \to b - \to c \to c - \to a] = 0$

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