Let $\overrightarrow{a}=\hat{i}+\hat{j};\overrightarrow{b}=2\hat{i}-\hat{k}$. Then vector $\overrightarrow{r}$satisffing the equations $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$ is

$\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}or(\overrightarrow{r}-\overrightarrow{b})\times \overrightarrow{a}=0$

$\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}or(\overrightarrow{r}-\overrightarrow{a})\times \overrightarrow{b}=0$

If $\overrightarrow{r}=x\widehat{i}+y\widehat{j}+z\widehat{k}$, then

$\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ x-2& y& z+1\\ 1& 1& 0\end{array}\right|=0and\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ x-1& y-1& z\\ 2& 0& -1\end{array}\right|=0$

$\begin{array}{l}\Rightarrow z+1=0,x-y=2\\ andy-1=0,x-1+2z=0\\ \Rightarrow x=3,y=1,z=-1\end{array}$