If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ are three non-zero vectors and $\widehat n$ is a unit vector perpendicular to $\overrightarrow c$ such that $\overrightarrow a = \alpha \overrightarrow b - \widehat n,(\alpha \ne 0)$ and $\overrightarrow b \,.\overrightarrow c = 12$, then $$\left| {\overrightarrow c \times (\overrightarrow a \times \overrightarrow b )} \right|$$ is equal to :

<p>$\widehat n = \alpha \overrightarrow b - \overrightarrow a$</p> <p>$$\overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } \right) = \left( {\overrightarrow c \,.\,\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow c \,.\,\overrightarrow a } \right)\overrightarrow b $$</p> <p>$$ = 12\overrightarrow a - \left( {\overrightarrow c \,.\,\left( {\alpha \overrightarrow b - \widehat n} \right)} \right)\overrightarrow b $$</p> <p>$= 12\overrightarrow a - (12\alpha - 0)\overrightarrow b$</p> <p>$= 12\left( {\overrightarrow a - \alpha \overrightarrow b } \right)$</p> <p>$\therefore$ $$\left| {\overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right| = 12$$</p>