Explanation

(a)
Let $\overrightarrow{\mathbf{a}} = x\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}} = 2\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 5\widehat{\mathbf{k}}$
$Sience,\ \ \ \frac{\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}}}{|\overrightarrow{\mathbf{b}}|} = \frac{1}{\sqrt{30}}$
$\Longrightarrow \ \frac{\left( x\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) \bullet \left( 2\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 5\widehat{\mathbf{k}} \right)}{\left| \sqrt{4 + 1 + 25} \right|} = \frac{1}{\sqrt{30}}$
$\Longrightarrow 2x - 1 + 5 = 1$
$\Longrightarrow x = - \frac{3}{2}$