Explanation

(d)
Let the unit vector be $\overrightarrow{\mathbf{r}} = x\widehat{\dot{\mathbf{i}}} + y\widehat{\dot{\mathbf{j}}} + z\widehat{\mathbf{k}}$
Since, $\overrightarrow{\mathbf{r}} \bullet \left( 3\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right) = 0\$and $\overrightarrow{\mathbf{r}} \bullet \left( 2\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}} \right) = 0$
$\Longrightarrow 3x + y + 2z = 0\ and\ 2x - 2y + 4z = 0$
On solving, we get $x = 1,y = - 1$ and $z = - 1$
$\therefore\ Required\ unit\ vector = \frac{\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}}{\sqrt{1^{2} + 1^{2} + 1^{2}}} = \frac{\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}}{\sqrt{3}}$