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If $2 \bar{i} + 4 \bar{j} - 5 \bar{k}, \bar{i} + \bar{j} + \bar{k}, \bar{j} + 2 \bar{k}$ are the position vectors of the vertices A,B,C of a triangle respectively, then a unit vector along the median through A is

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$\frac{1}{\sqrt{174}} (5 \bar{i} + 10 \bar{j} - 7 \bar{k})$

$\frac{1}{7} (3 \bar{i} + 6 \bar{j} - 2 \bar{k})$

$\frac{1}{\sqrt{66}} (\bar{i} + \bar{j} - 8 \bar{k})$

$\frac{1}{\sqrt{214}} (3 \bar{i} + 6 \bar{j} - 13 \bar{k})$

answer is B.

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Detailed Solution

$\bar{O D} = \frac{\bar{O B} + \bar{O C}}{2} = \frac{1}{2} \bar{i} + \bar{j} + \frac{3}{2} \bar{k}$
Median $\bar{A D} = \bar{O D} - \bar{O A} = \frac{- 1}{2} (3 \bar{i} + 6 \bar{j} - 13 \bar{k})$
Required vector=

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