Given lines are $\overrightarrow{r}=2\overrightarrow{b}+t\left(6\overrightarrow{c}-\overrightarrow{a}\right)$ and $\overrightarrow{r}=\overrightarrow{a}+s\left(\overrightarrow{b}-3\overrightarrow{c}\right)$

For point of intersection of the lines, the coefficient of $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ will be equal.

i.e. $2=s, 6t=-3s, t=-1$

Hence, the point of intersection will be $2\overrightarrow{b}-\left(6\overrightarrow{c}-\overrightarrow{a}\right)$