Explanation

(a)
$\ \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}\mathbf{=}\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}} \right)\mathbf{\bullet}\widehat{\mathbf{i}} = 1$
and $\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}\mathbf{=}\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}} \right)\mathbf{\bullet}\widehat{\mathbf{i}} = 1$
Now, $\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{c}}\mathbf{=}\left( \overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} \right)\overrightarrow{\mathbf{b}}\mathbf{-}\left( \overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{a}} = \mu\overrightarrow{\mathbf{b}} + \lambda\overrightarrow{\mathbf{a}}$
$\Rightarrow \mu = \overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{a}}\$ and $\lambda = - \overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}$
$\Rightarrow \ \mu = 1$ and $\lambda = - 1$
$\therefore\mu + \lambda = 1 - 1 = 0$