Explanation

(a)
Let $\overrightarrow{\mathbf{a}} = x\widehat{\dot{\mathbf{i}}} + y\widehat{\dot{\mathbf{j}}} + z\widehat{\mathbf{k}}$
$\therefore\ \left( \overrightarrow{\mathbf{a}} \bullet \widehat{\dot{\mathbf{i}}} \right)\widehat{\dot{\mathbf{i}}} = \left\lbrack \left( x\widehat{\dot{\mathbf{i}}} + y\widehat{\dot{\mathbf{j}}} + z\widehat{\mathbf{k}} \right) \bullet \widehat{\dot{\mathbf{i}}} \right\rbrack\widehat{\dot{\mathbf{i}}} = x\widehat{\dot{\mathbf{i}}}$
Similarly, $\left( \overrightarrow{\mathbf{a}} \bullet \widehat{\dot{\mathbf{j}}} \right)\widehat{\dot{\mathbf{i}}} = y\widehat{\dot{\mathbf{j}}},\left( \overrightarrow{\mathbf{a}} \bullet \widehat{\mathbf{k}} \right)z\widehat{\mathbf{k}}$
$\therefore\left( \overrightarrow{\mathbf{a}} \bullet \widehat{\dot{\mathbf{i}}} \right)\widehat{\dot{\mathbf{i}}} + \left( \overrightarrow{\mathbf{a}} \bullet \widehat{\dot{\mathbf{j}}} \right)\widehat{\dot{\mathbf{j}}} + \left( \overrightarrow{\mathbf{a}} \bullet \widehat{\mathbf{k}} \right)\widehat{\mathbf{k}} = x\widehat{\dot{\mathbf{i}}} + y\widehat{\dot{\mathbf{j}}} + z\widehat{\mathbf{k}} = \overrightarrow{\mathbf{a}}$