Let $\overrightarrow{\mathrm{u}}=\hat{i}-\widehat{\mathrm{j}}-2\widehat{\mathrm{k}},\overrightarrow{\mathrm{v}}=2\widehat{\mathrm{i}}+\widehat{\mathrm{j}}-\widehat{\mathrm{k}},\overrightarrow{\mathrm{v}}\cdot \overrightarrow{\mathrm{w}}=2\text{ and }\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{w}}=\overrightarrow{\mathrm{u}}+\mathrm{\lambda }\overrightarrow{\mathrm{v}}\text{. Then }\overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{w}}$ is equal to 

$\begin{array}{l}(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{w}})\times \overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{u}}\times \overrightarrow{\mathrm{v}}\\ \Rightarrow |\overrightarrow{\mathrm{v}}{|}^2\overrightarrow{\mathrm{w}}-(\overrightarrow{\mathrm{v}}\cdot \overrightarrow{\mathrm{w}})\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{u}}\times \overrightarrow{\mathrm{v}}\\ \Rightarrow \overrightarrow{\mathrm{w}}=\left(\frac{7}{6},\frac{-1}{6},\frac{1}{6}\right)\\ \overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{w}}=1\end{array}$

or

$$\begin{gathered} \begin{array}{l}\overrightarrow{\mathrm{u}}=(1,-1,-2),\overrightarrow{\mathrm{v}}=(2,1,-1),\overrightarrow{\mathrm{v}}\cdot \overrightarrow{\mathrm{w}}=2\\ \overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{w}}=\overrightarrow{\mathrm{u}}+\lambda \overrightarrow{\mathrm{v}}\dots \dots \dots \dots \dots \dots \dots \left(1\right)\\ \text{ Taking dot with }\overrightarrow{\mathrm{w}}\text{ in }\left(1\right)\\ \overrightarrow{\mathrm{w}}\cdot (\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{w}})=\overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{w}}+\lambda \overrightarrow{\mathrm{v}}\cdot \overrightarrow{\mathrm{w}}\\ \Rightarrow 0=\overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{w}}+2\lambda \end{array} \\ \begin{array}{r}\lambda =-\frac{1}{2}\\ \Rightarrow \overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{w}}=-2\lambda =1\end{array} \end{gathered}$$