Let u → = i ^ - j ^ - 2 k ^ ,   v → = 2 i ^ + j ^ - k ^ , v → · w → = 2…

Given:

$\overrightarrow{v}\times \overrightarrow{w}=\overrightarrow{u}+\lambda \overrightarrow{v} ....\left(1\right)$

Taking dot product with $\overrightarrow{w}$, we get

$\overrightarrow{w}·\left(\overrightarrow{v}\times \overrightarrow{w}\right)=\overrightarrow{w}·\overrightarrow{u}+\lambda \left(\overrightarrow{w}·\overrightarrow{v}\right)$

$\Rightarrow \overrightarrow{u}·\overrightarrow{w}+\lambda \left(\overrightarrow{w}·\overrightarrow{v}\right)=0$

$\Rightarrow \overrightarrow{u}·\overrightarrow{w}=-2\lambda ....\left(2\right)$

Taking dot product with $\overrightarrow{v}$ in $\left(1\right)$, we get

$\overrightarrow{v}·\left(\overrightarrow{v}\times \overrightarrow{w}\right)=\overrightarrow{v}·\overrightarrow{u}+\lambda \left(\overrightarrow{v}·\overrightarrow{v}\right)$

$\Rightarrow \left(2-1+2\right)+\lambda \left(6+1+1\right)=0$

$\Rightarrow \lambda =-\frac{1}{2}$

So, by $\left(2\right)$, we get

$\overrightarrow{u}·\overrightarrow{w}=-2\lambda =1$