Given, a → = 2 i ^ + j ^ - 2 k ^ and b → =   i ^ + j ^ . Let c → be a vector such…

Given,

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

$\overrightarrow{b}=\hat{i}+\hat{j}$

Now,

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

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

$\Rightarrow \left|\overrightarrow{a}\times \overrightarrow{b}\right|=3$

Given,

$\left|\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \overrightarrow{c}\right|=3$

$\Rightarrow \left|\overrightarrow{a}\times \overrightarrow{b}\right|\left|\overrightarrow{c}\right|\sin 30°=3$

$\Rightarrow \left|\overrightarrow{c}\right|=2$

Now,

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

$\Rightarrow {\left|\overrightarrow{c}-\overrightarrow{a}\right|}^2=9$

$\Rightarrow {\left|\overrightarrow{c}\right|}^2+{\left|\overrightarrow{a}\right|}^2-2\left(\overrightarrow{a}·\overrightarrow{c}\right)=9$

$\Rightarrow \overrightarrow{a}.\overrightarrow{c}=2$