Explanation

(b)
Let $\overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}}$ and $\overrightarrow{\mathbf{c}} = \widehat{\dot{\mathbf{j}}}$
$\therefore|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}| = |\widehat{\mathbf{k}}| = 1$
Let $\overrightarrow{\mathbf{a}} = a_{1}\widehat{\dot{\mathbf{i}}} + a_{2}\widehat{\dot{\mathbf{j}}} + a_{3}\widehat{\mathbf{k}}$
Now, $\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{a}}\ \widehat{\dot{\mathbf{i}}} = a_{1},\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{a}} \bullet \widehat{\dot{\mathbf{j}}} = a_{2}$
$and\ \ \ \overrightarrow{\mathbf{a}} \bullet \frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|} = \overrightarrow{\mathbf{a}} \bullet \widehat{\mathbf{k}} = a_{3}\ \$
$\therefore\left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} \right) + \overrightarrow{\mathbf{b}} + \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{c}} \right)\overrightarrow{\mathbf{c}} + \frac{\overrightarrow{\mathbf{a}} \bullet (\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})}{|\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}|} \bullet (\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})$
$= a_{1}\overrightarrow{\mathbf{b}} + a_{2}\overrightarrow{\mathbf{c}} + a_{3}\left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right)$
$= a_{1}\widehat{\dot{\mathbf{i}}} + a_{2}\widehat{\dot{\mathbf{j}}} + a_{3}\widehat{\mathbf{k}} = \overrightarrow{\mathbf{a}}$