Explanation

(d)
$We\ have,\ \overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$
$\therefore\ \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\dot{\mathbf{i}}} \right) + \widehat{\dot{\mathbf{j}}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\dot{\mathbf{j}}} \right) + \widehat{\mathbf{k}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\mathbf{k}} \right)\ \ldots\ldots(i)$
$Now,\ \ \widehat{\dot{\mathbf{i}}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\dot{\mathbf{i}}} \right) = \left( \widehat{\dot{\mathbf{i}}} \bullet \widehat{\dot{\mathbf{i}}} \right)\overrightarrow{\mathbf{a}} - \left( \widehat{\dot{\mathbf{i}}} \bullet \overrightarrow{\mathbf{a}} \right)\widehat{\dot{\mathbf{i}}}$
$= 1\left( \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}} \right) - (1)\widehat{\dot{\mathbf{i}}}$
$= 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$
$Similarly,\ \widehat{\dot{\mathbf{j}}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\dot{\mathbf{j}}} \right) = \widehat{\dot{\mathbf{i}}} + 3\widehat{\mathbf{k}}$
$and\ \widehat{\mathbf{k}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\mathbf{k}} \right) = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}}$
$\therefore From\ Eq.(i),$
$\overrightarrow{\mathbf{b}} = 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}} + \widehat{\dot{\mathbf{i}}} + 3\widehat{\mathbf{k}} + \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}}$
$= 2\widehat{\dot{\mathbf{i}}} + 4\widehat{\dot{\mathbf{j}}} + 6\widehat{\mathbf{k}}$
$\Longrightarrow \left| \overrightarrow{\mathbf{b}} \right| = \sqrt{4 + 16 + 36} = 2\sqrt{14}$