$ \Rightarrow |\overrightarrow c {|^2} + |\overrightarrow a {|^2} - 2\overrightarrow c \cdot \overrightarrow a = 14$ ........$(1)$
$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}+2|\overrightarrow{\mathrm{c}}|=0$
$\Rightarrow \quad|\overrightarrow{\mathrm{a}}| \cdot|\overrightarrow{\mathrm{c}}| \cdot \cos \theta+2|\overrightarrow{\mathrm{c}}|=0$
$ \Rightarrow |\overrightarrow c | \cdot (|\overrightarrow {\rm{a}} | \cdot \cos \theta + 2) = 0$
$\Rightarrow \quad \cos \theta=-\frac{2}{3},$ given $|\vec{a}|=3$
from $(i)$
$\Rightarrow \quad|\overrightarrow{\mathrm{c}}|^{2}+9-2|\overrightarrow{\mathrm{c}}| \cdot|\overrightarrow{\mathrm{a}}| \cdot\left(-\frac{2}{3}\right)-14=0$
$\Rightarrow \quad|\overrightarrow{\mathrm{c}}|^{2}+4|\overrightarrow{\mathrm{c}}|-5=0 \Rightarrow|\overrightarrow{\mathrm{c}}|=1,-5$
$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\left|\begin{array}{ccc}{\hat{\mathrm{i}}} & {\hat{\mathrm{j}}} & {\hat{\mathrm{k}}} \\ {2} & {1} & {-2} \\ {1} & {1} & {0}\end{array}\right|=2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$
$|(\vec{a} \times \vec{b}) \times \vec{c}|=|(\vec{a} \times \vec{b})| \cdot|\vec{c}| \cdot \sin \theta$
$=3.1 \times \frac{1}{2}=\frac{3}{2}$
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