Explanation

(a)
Let $\overrightarrow{\mathbf{a}} = x\widehat{\mathbf{i}} + y\widehat{\mathbf{j}} + \mathcal{z}\widehat{\mathbf{k}}$
$\therefore\overrightarrow{\mathbf{a}}\mathbf{\bullet \ }\widehat{\mathbf{i}} = \left( x\widehat{\mathbf{i}} + y\widehat{\mathbf{j}} + \mathcal{z}\widehat{\mathbf{k}} \right) \bullet \widehat{\mathbf{i}} = x$
$\overrightarrow{\mathbf{a}}\mathbf{\bullet}\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}} \right) = \left( x\widehat{\mathbf{i}} + y\widehat{\mathbf{j}} + \mathcal{z}\widehat{\mathbf{k}} \right) \bullet \left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}} \right) = x + y$
and $\overrightarrow{\mathbf{a}}\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}} \right) = \left( x\widehat{\mathbf{i}} + y\widehat{\mathbf{j}} + \mathcal{z}\widehat{\mathbf{k}} \right) \bullet \left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}} \right) = x + y + \mathcal{z}$
$\because$ Given that, $\overrightarrow{\mathbf{a}}\mathbf{\bullet}\widehat{\mathbf{i}}\mathbf{=}\overrightarrow{\mathbf{a}}\mathbf{\bullet}\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}} \right)\mathbf{=}\overrightarrow{\mathbf{a}}\mathbf{\bullet}\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}} \right)$
$\Rightarrow x = x + y = x + y + \mathcal{z}$
Take $x = x + y\ \Rightarrow y = 0$
and $x + y = x + y + \mathcal{z\ \Rightarrow \ z} = 0$
$\Rightarrow x$ has any real values.
Now, take $x = 1\ \therefore\ \overrightarrow{\mathbf{a}}\mathbf{=}\widehat{\mathbf{i}}$