If $a$ and $b$ are respectively the internal and external bisectors of the angles between the vectors
$- i + 2 j - 2 k$ and $3 i + 4 j$ and $| a | = \frac{2}{3} \sqrt{6}, | b | = \frac{2}{3} \sqrt{3},$ then one of the values of $a - b$ is

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$\frac{1}{10} (- 8 i + 11 j - 2 k)$

$\frac{2}{3} (- i + 2 j - 2 k)$

$\frac{1}{15} (9 i - 11 j + 3 k)$

$\frac{1}{12} (2 i - 3 j - k)$

answer is B.

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Detailed Solution

$\hat{c} = \frac{- i + 2 j - 2 k}{3} \hat{d} = \frac{3 i + 4 j}{5} \hat{c} + \hat{d} = \frac{4 i + 22 j - 10 k}{15} |\hat{c} + \hat{d}| = \frac{2}{15} (\sqrt{4 + 121 + 25}) \hat{a} = \frac{2}{3} \sqrt{6} (\frac{2 i + 11 j - 5 k}{\sqrt{150}}) = \frac{2}{15} (2 i + 11 j - 5 k) \hat{c} - \hat{d} = - 14 i - 2 j - 10 k = - 2 (7 i + j + 5 j) |\hat{c} - \hat{d}| = 2 \sqrt{49 + 1 + 25} = 2 \times 5 \sqrt{3} \hat{b} = \frac{2}{3} \sqrt{3} (\frac{- 7 i - 3 - 5 k}{5 \sqrt{3}}) = \frac{2}{15} (- 7 i - j - 5 k)$