Let $\overrightarrow{a}=3\widehat{i}+\widehat{j}$ and $\overrightarrow{b}=\widehat{i}+2\widehat{j}+\widehat{k}$ . Let $\overrightarrow{c}$  be a vector satisfying $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{b}+\lambda \overrightarrow{c}$ . If $\overrightarrow{b}$  and   $\overrightarrow{c}$ are non-parallel, then the value of  $\lambda$ is:

Here we are given that 
 $\overrightarrow{a}=3\widehat{i}+\widehat{j},\overrightarrow{b}=\widehat{i}+2\widehat{j}+\widehat{k}$
As  $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{b}+\lambda \overrightarrow{c}$
$$\begin{gathered} =\overrightarrow{a}.\overrightarrow{c}\left(\overrightarrow{b}\right)-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c}=\overrightarrow{b}+\lambda \overrightarrow{c} \\ \Rightarrow \overrightarrow{a}.\overrightarrow{c}=1,\overrightarrow{a}.\overrightarrow{b}=-\lambda \\ \Rightarrow (3\widehat{i}+\widehat{j}).(\widehat{i}+2\widehat{j}+\widehat{k})=-\lambda \\ \Rightarrow \lambda =-5 \end{gathered}$$