Let $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}.$ Let a vector $\overrightarrow{v}$ be in the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}.$ If $\overrightarrow{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\overrightarrow{a}$ is $19$ units, then $|2\overrightarrow{v}{|}^2$ is equal to _____.

$\overrightarrow{v},\overrightarrow{a},\overrightarrow{b}$ are coplanar

$\overrightarrow{v}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}$
$\overrightarrow{v}=\left(2\mathrm{\lambda }+\mu \right)\hat{i}+\left(2\mu -\lambda \right)\hat{j}+\left(2\lambda -\mu \right)\hat{k}$

given $\overrightarrow{v}\perp$$\left(3\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{k}\right)$

$\overrightarrow{v}·\left(3\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{k}\right)=0$

$\Rightarrow (\left(2\mathrm{\lambda }+\mu \right)\hat{i}+\left(2\mu -\lambda \right)\hat{j}+\left(2\lambda -\mu \right)\hat{k}).(3\hat{i}+2\hat{j}-\hat{k})=0$

$\Rightarrow 3(2\lambda +\mu )+2(2\mu -\lambda )-(2\lambda -\mu )=0$

$\Rightarrow \lambda =-4\mathrm{\mu } ...\left(\mathrm{i}\right)$

and also $\overrightarrow{v}·\hat{a}=19\Rightarrow 9\mathrm{\lambda }-2\mathrm{\mu }=57 ...\left(\mathrm{ii}\right)$

solving $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$ we get

$\lambda =6, \mu =-\frac{3}{2}$
$\Rightarrow 2\overrightarrow{\mathrm{v}}=21\hat{\mathrm{i}}-18\hat{\mathrm{j}}+27\hat{\mathrm{k}}$
$\Rightarrow |2\overrightarrow{\mathrm{v}}{|}^2=441+324+729$
$=1494$