Given,

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

 Let $\overrightarrow{c}$ be a vector satisfying $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\overrightarrow{b}+\lambda \overrightarrow{c}$. If $\overrightarrow{b}$ and $\overrightarrow{c}$ are non-parallel, then the value of $\lambda$ is

Now putting the value of $\overrightarrow{a}=3\hat{i}+\hat{j} \& \overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$ in  $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\overrightarrow{b}+\lambda \overrightarrow{c}$ we get,

$\Rightarrow \overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\overrightarrow{b}+\lambda \overrightarrow{c}$

$\Rightarrow \overrightarrow{a}·\overrightarrow{c}\left(\overrightarrow{b}\right)-\left(\overrightarrow{a}·\overrightarrow{b}\right)\overrightarrow{c}=\overrightarrow{b}+\lambda \overrightarrow{c}$

On comparing both side we get,

$\Rightarrow \overrightarrow{a}·\overrightarrow{c}=1 \& \overrightarrow{a}·\overrightarrow{b}=-\lambda$

$\Rightarrow \left(3\hat{i}+\hat{j}\right)·\left(\hat{i}+2\hat{j}+\hat{k}\right)=-\lambda$

$\Rightarrow \lambda =-5$