Let three vectors $\overrightarrow a ,\overrightarrow b$ and $\overrightarrow c$ be such that $\overrightarrow c$ is coplanar
with $\overrightarrow a$ and $\overrightarrow b$, $\overrightarrow a .\overrightarrow c$ = 7 and $\overrightarrow b$ is perpendicular to $\overrightarrow c$, where
$\overrightarrow a = - \widehat i + \widehat j + \widehat k$ and $\overrightarrow b = 2\widehat i + \widehat k$ , then the
value of $$2{\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2}$$ is _____.

$$\overrightarrow c = \lambda (\overrightarrow b \times (\overrightarrow a \times \overrightarrow b ))$$<br><br>$$ = \lambda ((\overrightarrow b \,.\,\overrightarrow b )\overrightarrow a - (\overrightarrow b \,.\,\overrightarrow a )\overrightarrow b )$$<br><br>$$ = \lambda (5( - \widehat i + \widehat j + \widehat k) + 2\widehat i + \widehat k)$$<br><br>$= \lambda ( - 3\widehat i + 5\widehat j + 6\widehat k)$<br><br>$\overrightarrow c \,.\,\overrightarrow a = 7$ <br/><br/>$\Rightarrow 3\lambda + 5\lambda + 6\lambda = 7$<br><br>$\Rightarrow$ $\lambda = {1 \over 2}$<br><br>$\therefore$ $$2{\left| {\left( {{{ - 3} \over 2} - 1 + 2} \right)\widehat i + \left( {{5 \over 2} + 1} \right)\widehat j + (3 + 1 + 1)\widehat k} \right|^2}$$<br><br>$= 2\left( {{1 \over 4} + {{49} \over 4} + 25} \right) = 25 + 50 = 75$