Explanation

(c)
Let $\overrightarrow{\mathbf{a}} = a_{1}\widehat{\dot{\mathbf{i}}} + a_{2}\widehat{\dot{\mathbf{j}}} + a_{3}\widehat{\mathbf{k}}$
Given, $\overrightarrow{\mathbf{a}} \bullet \widehat{\dot{\mathbf{i}}} = \overrightarrow{\mathbf{a}} \bullet \left( \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} \right) = \overrightarrow{\mathbf{a}} \bullet \left( \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) = 1$
$\therefore a_{1} = a_{1} + a_{2} = a_{1} + a_{2} + a_{3} = 1$
$\Longrightarrow \ a_{1} = 1,\ a_{2} = 0,\ a_{3} = 0$
$\therefore\ \overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}}$