Explanation

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