Explanation

(a)
Given, $\overrightarrow{\mathbf{u}}\mathbf{\times}\overrightarrow{\mathbf{v}}\mathbf{+}\overrightarrow{\mathbf{u}}\mathbf{=}\overrightarrow{\mathbf{w}}$ and $\overrightarrow{\mathbf{w}}\mathbf{\times}\overrightarrow{\mathbf{u}}\mathbf{=}\overrightarrow{\mathbf{v}}$
$\Rightarrow \left( \mathbf{\ }\overrightarrow{\mathbf{u}}\mathbf{\times}\overrightarrow{\mathbf{v}}\mathbf{+}\overrightarrow{\mathbf{u}}\mathbf{\ } \right)\mathbf{\times}\overrightarrow{\mathbf{u}}\mathbf{=}\overrightarrow{\mathbf{v}}$
$\Rightarrow (\overrightarrow{\mathbf{u}}\mathbf{\times}\overrightarrow{\mathbf{v}}\mathbf{)\ \times}\overrightarrow{\mathbf{u}}\mathbf{=}\overrightarrow{\mathbf{v}}$
$\Rightarrow \ \overrightarrow{\mathbf{v}}\mathbf{-}\left( \overrightarrow{\mathbf{u}}\mathbf{\bullet}\overrightarrow{\mathbf{v}} \right)\mathbf{=}\overrightarrow{\mathbf{v}}$
$\Rightarrow \ \left( \overrightarrow{\mathbf{u}}\mathbf{\bullet}\overrightarrow{\mathbf{v}} \right)\overrightarrow{\mathbf{u}} = 0$
$\Rightarrow \ \left( \overrightarrow{\mathbf{u}}\mathbf{\bullet}\overrightarrow{\mathbf{v}} \right) = 0$
Now, $\left\lbrack \mathbf{\ }\overrightarrow{\mathbf{u}}\mathbf{\ }\overrightarrow{\mathbf{v}}\mathbf{\ }\overrightarrow{\mathbf{w}}\mathbf{\ } \right\rbrack\mathbf{=}\overrightarrow{\mathbf{u}}\mathbf{\bullet (}\overrightarrow{\mathbf{v}}\mathbf{\times \ }\overrightarrow{\mathbf{w}})$
$= \overrightarrow{\mathbf{u}}\mathbf{\bullet}\left( \mathbf{\ }\overrightarrow{\mathbf{v}}\mathbf{\times}\left( \mathbf{\ }\overrightarrow{\mathbf{u}}\mathbf{\times}\overrightarrow{\mathbf{v}}\mathbf{+}\overrightarrow{\mathbf{u}}\mathbf{\ } \right)\mathbf{\ } \right)$
$= \overrightarrow{\mathbf{u}}\mathbf{\bullet (}\overrightarrow{\mathbf{v}}\mathbf{\ }\left( \mathbf{\ }\overrightarrow{\mathbf{u}}\mathbf{\times}\overrightarrow{\mathbf{v}}\mathbf{) +}\overrightarrow{\mathbf{v}}\mathbf{+}\overrightarrow{\mathbf{u}}\mathbf{\ } \right)$
$= \overrightarrow{\mathbf{u}}\mathbf{\bullet (}{\overrightarrow{\mathbf{v}}}^{\mathbf{2}}\mathbf{\times}\overrightarrow{\mathbf{u}}\mathbf{- (}\overrightarrow{\mathbf{u}}\mathbf{\bullet}\overrightarrow{\mathbf{v}}\mathbf{) \bullet}\overrightarrow{\mathbf{v}}\mathbf{+}\overrightarrow{\mathbf{v}}\mathbf{\times}\overrightarrow{\mathbf{u}}$
$= {\overrightarrow{\mathbf{v}}}^{\mathbf{2}}{\overrightarrow{\mathbf{u}}}^{\mathbf{2}} = 1$