Let $\overrightarrow a = \widehat i + \widehat j + \widehat k$ and $\overrightarrow b = \widehat j - \widehat k$. If $\overrightarrow c$ is a vector such that $\overrightarrow a \times \overrightarrow c = \overrightarrow b$ and $\overrightarrow a .\overrightarrow c = 3$, then $\overrightarrow a .(\overrightarrow b \times \overrightarrow c )$ is equal to :

$\left| {\overrightarrow a } \right| = \sqrt 3$; $\overrightarrow a .\overrightarrow c = 3$; $$\overrightarrow a \times \overrightarrow b = - 2\widehat i + \widehat j + \widehat k$$, $\overrightarrow a \times \overrightarrow c = \overrightarrow b$<br><br>Cross with $\overrightarrow a$,<br><br>$$\overrightarrow a \times (\overrightarrow a \times \overrightarrow c ) = \overrightarrow a \times \overrightarrow b $$<br><br>$$ \Rightarrow (\overrightarrow a .\overrightarrow c )\overrightarrow a - {a^2}\overrightarrow c = \overrightarrow a \times \overrightarrow b $$<br><br>$$ \Rightarrow 3\overrightarrow a - 3\overrightarrow c = - 2\widehat i + \widehat j + \widehat k$$<br><br>$$ \Rightarrow 3\widehat i + 3\widehat j + 3\widehat k - 3\overrightarrow c = - 2\widehat i + \widehat j + \widehat k$$<br><br>$$ \Rightarrow \overrightarrow c = {{5\widehat i} \over 3} + {{2\widehat j} \over 3} + {{2\widehat k} \over 3}$$<br><br>$\therefore$ $$\overrightarrow a .(\overrightarrow b \times \overrightarrow c ) = (\overrightarrow a \times \overrightarrow b ).\overrightarrow c = {{ - 10} \over 3} + {2 \over 3} + {2 \over 3} = - 2$$