Let a → = 2 i ^ + 7 j ^ - k ^ ,   b ^ = 3 i ^ + 5 k ^ and c → = i ^ - j ^ + 2 k ^ Let d…

Given,

$\overrightarrow{a}=2\hat{i}+7\hat{j}-\hat{k}, \hat{b}=3\hat{i}+5\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+2\hat{k}$ 

And  $\overrightarrow{d}$ be a vector which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$,

So, $\overrightarrow{d}=\lambda \left(\overrightarrow{a}\times \overrightarrow{b}\right)$

Now finding, $\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 7& -1\\ 3& 0& 5\end{array}\right|$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=35\hat{i}-13\hat{j}-21\hat{k}$

So, $\overrightarrow{d}=\lambda \left(35\hat{i}-13\hat{j}-21\hat{k}\right)$

Also given, 

$\overrightarrow{c}·\overrightarrow{d}=12$

$\Rightarrow \overrightarrow{c}·\overrightarrow{d}=\left(\hat{i}-\hat{j}+2\hat{k}\right)·\lambda \left(35\hat{i}-13\hat{j}-21\hat{k}\right)$

$\Rightarrow \lambda (35+13-42)=12$

$\Rightarrow \lambda = 2$

So, $\overrightarrow{c}\times \overrightarrow{d}=2\times \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -1& 2\\ 35& -13& -21\end{array}\right|$

$\Rightarrow \overrightarrow{c}\times \overrightarrow{d}=2\left(47\hat{i}+91\hat{j}+22\hat{k}\right)$

Now finding,

$\left(-\hat{i}+\hat{j}-\hat{k}\right)·\left(\overrightarrow{c}\times \overrightarrow{d}\right)=2\left(-\hat{i}+\hat{j}-\hat{k}\right)\left(47\hat{i}+91\hat{j}+22\hat{k}\right)$

$\Rightarrow \left(-\hat{i}+\hat{j}-\hat{k}\right)·\left(\overrightarrow{c}\times \overrightarrow{d}\right)=2\left(-47+91-22\right)=44$