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The unit vector which is orthogonal to the vector $3 i + 2 j + 6 k$ and is coplanar with the vectors $2 i + j + k$ and $i - j + k$ is

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$\frac{2 \bar{i} - 6 \bar{j} + \bar{k}}{\sqrt{41}}$

$\frac{2 \bar{i} - 3 \bar{j}}{\sqrt{13}}$

$\frac{3 \bar{j} - \bar{k}}{\sqrt{10}}$

$\frac{4 \bar{i} + 3 \bar{j} - 3 \bar{k}}{\sqrt{34}}$

answer is C.

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Detailed Solution

$Let \bar{c} = \frac{3 \bar{j} - \bar{k}}{\sqrt{10}} \Rightarrow | \bar{c} | = 1 Let \bar{a} = 2 i + \bar{j} + \bar{k}$

$\Rightarrow \bar{b} = \bar{i} - \bar{j} + \bar{k}, \bar{d} = 3 \bar{i} + 2 \bar{j} + 6 \bar{k}$

$\Rightarrow [\bar{a} \bar{b} \bar{c}] = 0 and \bar{c} \cdot \bar{d} = 0$

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