Given, $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$$$

$\therefore \left(\overrightarrow{r}-\overrightarrow{b}\right)\times \overrightarrow{a}=0$

The Equation of line is, $\overrightarrow{r}=\overrightarrow{b}+p\overrightarrow{a}$

Similarly, for second line $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$

The Equation of line is $\overrightarrow{r}=\overrightarrow{b}+q\overrightarrow{a}$

As they both intersect $\overrightarrow{b}+q\overrightarrow{a}=\overrightarrow{b}+p\overrightarrow{a}$

On comparing the coefficients we get $p=q=1$

Hence, the equation of line is $\overrightarrow{r}=\overrightarrow{b}+\overrightarrow{a}$

$\Rightarrow \overrightarrow{r}=\hat{i}+\hat{j}$