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Scalar Triple Product

If a,b,c are ...

Question

If $\to a, \to b, \to c$ are three non-coplanar vectors such that volume of parallelopiped formed with $\to a, \to b, \to c$ as coterminous edges is equal to volume of parallelopiped formed with $\to a \times \to b, \to b \times \to c, \to c \times \to a$ as coterminous edges, then which of the following is/are correct $?$

A

[\to a \to b \to c] = 2

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B

[\to a \to b \to c] = - 1

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C

[\to a \to b \to c] = 0

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D

[\to a \to b \to c] = 1

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Solution

The correct option is D $[\to a \to b \to c] = 1$
As $\to a, \to b, \to c$ are three non-coplanar vector
So, $[\to a \to b \to c] \neq 0$
Also Volume of parallelopiped whose coterminous sides are $\to a, \to b, \to c$ is $= | [\to a \to b \to c] |$ and Volume of parallelopiped whose coterminous sides are $(\to a \times \to b), (\to b \times \to c), (\to c \times \to a)$ is $= | [\to a \times \to b \to b \times \to c \to c \times \to a] |$
Given, $| [\to a \to b \to c] | = | [\to a \times \to b \to b \times \to c \to c \times \to a] | \dots (i)$
We know, $[\to a \times \to b \to b \times \to c \to c \times \to a] = [\to a \to b \to c]^{2}$
Putting in equation $(i)$
$| [\to a \to b \to c] | = | [\to a \to b \to c]^{2} |$
$[\to a \to b \to c] = \pm 1$

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