$[\vec{a} \times \vec{b} \vec{c} \times \vec{d} \vec{e} \times \vec{f}]$ is equal to

Moderate

A

$[\vec{a} \vec{b} \vec{d}] [\vec{c} \vec{e} \vec{f}] - [\vec{a} \vec{b} \vec{c}] [\vec{d} \vec{e} \vec{f}]$
B

$[\vec{a} \vec{b} \vec{e}] [\vec{f} \vec{c} \vec{d}] - [\vec{a} \vec{b} \vec{f}] [\vec{e} \vec{c} \vec{d}]$
C

$[\vec{c} \vec{d} \vec{a}] [\vec{b} \vec{e} \vec{f}] - [\vec{a} \vec{d} \vec{b}] [\vec{a} \vec{e} \vec{f}]$
D

$[\vec{a} \vec{c} \vec{e}] [\vec{b} \vec{d} \vec{f}]$

Solution

$Let \vec{A} = \vec{a} \times \vec{b}, \vec{B} = \vec{c} \times \vec{d} and \vec{C} = \vec{e} \times \vec{f}$
$We know that \vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) = \vec{C} \cdot (\vec{A} \times \vec{B})$
$\begin{array}{l} = (\vec{a} \times \vec{b}) \cdot [(\vec{c} \times \vec{d}) \times (\vec{e} \times \vec{f})] \\ = (\vec{a} \times \vec{b}) \cdot [{(\vec{c} \times \vec{d}) \cdot \vec{f}} \vec{e} - {(\vec{c} \times \vec{d}) \cdot \vec{e}} \vec{f}} \\ = [\vec{c} \vec{d} \vec{f}] [\vec{a} \vec{b} \vec{e}] - [\vec{c} \vec{d} \vec{e}] [\vec{a} \vec{b} \vec{f}] \end{array}$
Similarly, other parts can be obtained.

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