Explanation

(c)
Given that, $\overrightarrow{\mathbf{a}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}}\mathbf{,\ \ }\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = 1$ and $\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}\mathbf{=}\widehat{\mathbf{j}}\mathbf{-}\widehat{k}$
As we know $\overrightarrow{\mathbf{a}}\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right)\mathbf{=}\left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{a}}\mathbf{-}\left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{b}}$
$\Rightarrow \left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}} \right)\mathbf{\times}\left( \widehat{\mathbf{j}}\mathbf{-}\widehat{\mathbf{k}} \right)\mathbf{= \ }\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}} \right) - \left( \sqrt{3} \right)^{2}\overrightarrow{\mathbf{b}}$
$\Rightarrow \ - 2\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}}\mathbf{-}3\overrightarrow{\mathbf{b}}$
$\mathbf{\Rightarrow \ }3\overrightarrow{\mathbf{b}}\mathbf{=}3\widehat{\mathbf{i}}$
$\Rightarrow \ \overrightarrow{\mathbf{b}}\mathbf{=}\widehat{\mathbf{i}}$