Let a → = 2 i ^ + j ^ - 2 k ^ and b → = i ^ + j ^ . If c → is a vector such that a →…

We have,

$\left|\overrightarrow{a}\right|=3=a$

$\overrightarrow{a}·\overrightarrow{c}=\left|\overrightarrow{c}\right|=c$

Now,

$\left|\overrightarrow{c}-\overrightarrow{a}\right|=2\sqrt{2}$

$\Rightarrow {\left|\overrightarrow{c}-\overrightarrow{a}\right|}^2=8$

$\Rightarrow c^2+a^2-2\overrightarrow{c}·\overrightarrow{a}=8$

$\Rightarrow c^2+9-2\left(c\right)=8$

$\Rightarrow c^2-2c+1=0$

$\Rightarrow {\left(c-1\right)}^2=0$

$\Rightarrow c=1=\left|\overrightarrow{c}\right|$

Now,

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 1& -2\\ 1& 1& 0\end{array}\right|$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=2\hat{i}-2\hat{j}+\hat{k}$

$\Rightarrow \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{2^2+{\left(-2\right)}^2+1^2}=3$

Now,

$\left|\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \overrightarrow{c}\right|=\left|\overrightarrow{a}\times \overrightarrow{b}\right|\left|\overrightarrow{c}\right|\sin \left(\frac{\pi }{6}\right)$

$\Rightarrow \left|\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \overrightarrow{c}\right|=3·1·\frac{1}{2}=\frac{3}{2}$