Explanation

(b)
Let $\overrightarrow{\mathbf{a}} = a_{1}\widehat{\dot{\mathbf{i}}} + a_{2}\widehat{\dot{\mathbf{j}}} + a_{3}\widehat{\mathbf{k}}$
$\therefore\overrightarrow{\mathbf{u}} = \widehat{\dot{\mathbf{i}}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\dot{\mathbf{i}}} \right) + \widehat{\dot{\mathbf{j}}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\dot{\mathbf{j}}} \right) + \widehat{\mathbf{k}} \times \left( \overrightarrow{\mathbf{a}} \times \widehat{\mathbf{k}} \right)$
$= \ \widehat{\dot{\mathbf{i}}} \times \left( - a_{2}\widehat{\mathbf{k}} + a_{3}\widehat{\dot{\mathbf{j}}} \right) + \widehat{\dot{\mathbf{j}}} \times \left( a_{1}\widehat{\mathbf{k}} - a_{3}\widehat{\dot{\mathbf{i}}} \right) + \widehat{\mathbf{k}} \times \left( - a_{1}\widehat{\dot{\mathbf{j}}} + a_{2}\widehat{\dot{\mathbf{i}}} \right)$
$= a_{2}\widehat{\dot{\mathbf{j}}} + a_{3}\widehat{\mathbf{k}} + a_{1}\widehat{\dot{\mathbf{i}}} + a_{3}\widehat{\mathbf{k}} + a_{1}\widehat{\dot{\mathbf{i}}} + a_{2}\widehat{\dot{\mathbf{j}}}$
$= 2\overrightarrow{\mathbf{a}}$