Explanation

(b)
Let $\overrightarrow{\mathbf{r}} = \left( 3\widehat{\dot{\mathbf{i}}} + 4\widehat{\dot{\mathbf{j}}} + 5\widehat{\mathbf{k}} \right) + b(6\widehat{\dot{\mathbf{i}}} - 7\widehat{\dot{\mathbf{j}}} - 3\widehat{\mathbf{k}})$
$= (3 + 6b)\widehat{\dot{\mathbf{i}}} + (4 - 7b)\widehat{\dot{\mathbf{j}}} + (5 - 3b)\widehat{\mathbf{k}}$
Since, $\overrightarrow{\mathbf{r}} \bullet \left( \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right) = 0$
$\Longrightarrow (3 + 6b)1 + (4 - 7b)1 - (5 - 3b)1 = 0$
$\Longrightarrow b = - 1$
$\therefore\ \overrightarrow{\mathbf{r}} = - 3\widehat{\dot{\mathbf{i}}} + 11\widehat{\dot{\mathbf{j}}} + 8\widehat{\mathbf{k}}$