Explanation

(c)
$\widehat{\mathbf{u}} \bullet \widehat{\mathbf{v}} = 0$
$\Longrightarrow \left| \widehat{\mathbf{u}} \right|\left| \widehat{\mathbf{v}} \right|\cos{\theta = 0}$
$\Longrightarrow 1 \times 1 \times \cos\theta = 0$ $(\because\left| \widehat{\mathbf{u}} \right| = \left| \widehat{\mathbf{v}} \right| = 1)$
$\Longrightarrow \cos\theta = 0$
$\Longrightarrow \ \theta = 90{^\circ}$
Let $\widehat{\mathbf{n}}$ be a unit vector perpendicular to the plane of vectors $\widehat{\mathbf{u}}$ and $\widehat{\mathbf{v}}$.
$\Longrightarrow \ \widehat{\mathbf{u}} \times \widehat{\mathbf{v}} = \left| \widehat{\mathbf{u}} \right|\left| \widehat{\mathbf{v}} \right|\sin{90{^\circ}\ \bullet \widehat{\mathbf{n}} = \widehat{\mathbf{n}}}$
Since, $\overrightarrow{\mathbf{r}}$ is coplanar with $\widehat{\mathbf{u}}$ and $\widehat{\mathbf{v}}$
$\therefore\ \widehat{\mathbf{n}}$ is perpendicular to $\overrightarrow{\mathbf{r}}$
Let $⏀$ be the angle between $\widehat{\mathbf{n}}\$ and $\overrightarrow{\mathbf{r}}$
$\Longrightarrow \ ⏀ = 90{^\circ}$
$\therefore\left| \overrightarrow{\mathbf{r}} \times \left( \widehat{\mathbf{u}} \times \widehat{\mathbf{v}} \right) \right)\left| = \left| \overrightarrow{\mathbf{r}} \times \widehat{\mathbf{n}} \right| = \left| \overrightarrow{\mathbf{r}} \right| \right|\widehat{\mathbf{n}}|\sin ⏀$
$= \left| \overrightarrow{\mathbf{r}} \right| \times 1 \times \sin{90{^\circ}}$
$= |\overrightarrow{\mathbf{r}}|$