Explanation

(a)
We know that, any vector $\overrightarrow{\mathbf{a}}$ can be uniquely expressed in terms of three non-coplanar vectors as $\overrightarrow{\mathbf{a}} = x\widehat{\mathbf{i}} + y\widehat{\mathbf{j}}\mathcal{+ z}\widehat{\mathbf{k}}$ multiply in succession by $\widehat{\mathbf{i}}\mathbf{,}\widehat{\mathbf{\ j}}\ and\ \widehat{\mathbf{k}}$, we get
$x = \overrightarrow{\mathbf{a}}\mathbf{\bullet}\widehat{\mathbf{i}},\ y = \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{j}}\mathcal{,\ z =}\overrightarrow{\mathbf{a}}\mathbf{\bullet}\widehat{\mathbf{k}}$
$\therefore\left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\widehat{\mathbf{i}} \right)\widehat{\mathbf{i}} + \left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\widehat{\mathbf{j}} \right)\widehat{\mathbf{j}} + \left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\widehat{\mathbf{k}} \right)\widehat{\mathbf{k}} = x\widehat{\mathbf{i}} + y\widehat{\mathbf{j}}\mathcal{+ z}\widehat{\mathbf{k}}\mathbf{=}\overrightarrow{\mathbf{a}}$