Let, $\mathbf{O A}=2 \mathbf{a}+3 \mathbf{b}-\mathbf{c}, \mathbf{O B}=3 \mathbf{a}+4 \mathbf{b}-2 \mathbf{c}$
$$
\mathbf{O C}=\mathbf{a}-2 \mathbf{b}+3 \mathbf{c} \text {, and } \mathbf{O D}=\mathbf{a}-6 \mathbf{b}+6 \mathbf{c}
$$
The vector equation of line joining the points $A$ and $B$ is
$$
\begin{aligned}
\mathbf{r} & =\mathbf{O A}+t(\mathbf{O B}-\mathbf{O A}), t \in R \\
& =(2 \mathbf{a}+3 \mathbf{b}-\mathbf{c})+t(3 \mathbf{a}+4 \mathbf{b}-2 \mathbf{c})-(2 \mathbf{a}+3 \mathbf{b}-\mathbf{c})]
\end{aligned}
$$


Vector equation of the line joining the points $C$ and $D$ is
$$
\begin{aligned}
\mathbf{r} & =\mathbf{O C}+s(\mathbf{O D}-\mathbf{O C}) \\
& =\mathbf{a}-2 \mathbf{b}+3 \mathbf{c}+s(\mathbf{a}-6 \mathbf{b}+6 \mathbf{c})-(\mathbf{a}-2 \mathbf{b}+3 \mathbf{c})
\end{aligned}
$$

Comparing coefficient of a in Eq. (ii) and (iii)
we get $2+t=1 \Rightarrow t=-1$
Put in Eq. (i)
$$
\begin{aligned}
\mathbf{r} & =2 \mathbf{a}+3 \mathbf{b}-\mathbf{c}+(-1)[\mathbf{a}+\mathbf{b}-\mathbf{c}] \\
& =2 \mathbf{a}+3 \mathbf{b}-\mathbf{c}-\mathbf{a}-\mathbf{b}+\mathbf{c} \\
\mathbf{r} & =\mathbf{a}+2 \mathbf{b}
\end{aligned}
$$
So, the point of in intersection is $\mathbf{a}+2 \mathbf{b}$