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If $\bar{A} = \hat{i} + \hat{j} + 2 \hat{k}, \bar{B} = 2 \hat{i} + \hat{j} + \hat{k}, \bar{C} = 3 \hat{i} + \hat{j} + 2 \hat{k}$ and $\bar{D} = 4 \hat{i} + \hat{j} + 3 \hat{k}$ are vectors. Then, match list-I with directional cosines in List-II and choose correct option
List-I List-II
I) For $\bar{A}$ P) $(\frac{4}{\sqrt{26}}, \frac{1}{\sqrt{26}}, \frac{3}{\sqrt{26}})$
II) For $\bar{B}$ Q) $(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}})$
III) For $\bar{C}$ R) $(\frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{8}{\sqrt{14}})$
IV) For $\bar{D}$ S) $(\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}})$
T) $(\frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}})$

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$I \to S, I I \to R, I I I \to Q, I V \to P$

$I \to S, I I \to T, I I I \to P, I V \to Q$

$I \to Q, I I \to S, I I I \to R, I V \to T$

$I \to Q, I I \to S, I I I \to T, I V \to P$

answer is D.

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Detailed Solution

$| \bar{A} | = \sqrt{1 + 1 + 4} = \sqrt{6} (\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}})$
$| \bar{B} | = \sqrt{4 + 1 + 1} = \sqrt{6} (\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}})$
$| \bar{C} | = \sqrt{9 + 1 + 4} = \sqrt{14} (\frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}})$
$| \bar{D} | = \sqrt{16 + 1 + 9} = \sqrt{26} (\frac{4}{\sqrt{26}}, \frac{1}{\sqrt{26}}, \frac{3}{\sqrt{26}})$

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