Given: 

$\overrightarrow{a}=-5\hat{i}+j-3\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}-4\hat{k}$

$\Rightarrow \left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \hat{i}=\left(\overrightarrow{a}\cdot \hat{i}\right)\overrightarrow{b}-\left(\overrightarrow{b}\cdot \hat{i}\right)\overrightarrow{a}$

$\Rightarrow \left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \hat{i}=-5\overrightarrow{b}-\overrightarrow{a}$

$\Rightarrow \left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \hat{i}=\left(\left(\left(-5\overrightarrow{b}-\overrightarrow{a}\right)\times \hat{i}\right)\times \hat{i}\right)$

$\Rightarrow \left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \hat{i}=\left(\left(-11\hat{j}+23\hat{k}\right)\times \hat{i}\right)\times \hat{i}$

$\Rightarrow \left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \hat{i}=\left(11\hat{k}+23\hat{j}\right)\times \hat{i}$

$\Rightarrow \left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \hat{i}=\left(11\hat{j}-23\hat{k}\right)$

$\Rightarrow \overrightarrow{c}.\left(-\hat{i}+\hat{j}+\hat{k}\right)=11-23$

$\Rightarrow \overrightarrow{c}.\left(-\hat{i}+\hat{j}+\hat{k}\right)=-12$