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The minimum volume of the parallalopiped formed by the vectors $i + a j + k, j + a k$ and $a i + k$ is

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$\frac{1}{3 \sqrt{3}}$

$\frac{1}{\sqrt{3}}$

$\frac{3 \sqrt{3} - 2}{3 \sqrt{3}}$

$\frac{3 \sqrt{3} + 2}{3 \sqrt{3}}$

answer is B.

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Detailed Solution

$[a b c] = |\begin{matrix} 1 & a & 1 \\ 0 & 1 & a \\ a & 0 & 1 \end{matrix}| = 1 (1) - a (- a^{2}) + 1 (0 - a) f (a) = 1 + a^{3} - a f^{1} (a) = 3 a^{2} - 1 = 0 \Rightarrow a = \frac{1}{\sqrt{3}}$

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