Explanation

(c)
$\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{a}} = 1 + 1 + 1 = 3$
Using,
$\overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) = \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{a}} - \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{a}} \right)\overrightarrow{\mathbf{b}}$
$\therefore\left( \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) \times \left( \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right) = \left( \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) - 3\overrightarrow{\mathbf{b}}$
$\Longrightarrow \ - 2\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} - 3\overrightarrow{\mathbf{b}}$
$\Longrightarrow \ \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}}$