The vector equation of any plane passing through the line of intersection of the planes $\vec{r} \cdot \vec{m}_1=q_1$ and $\vec{r} \cdot \vec{m}_2=q_2$ is given by $\vec{r} \cdot (\vec{m}_1+\lambda \vec{m}_2)=q_1+\lambda q_2$ for $\lambda \in R$. Find the vector equation of the plane passing through the point $2 \hat{i}-3 \hat{j}+\hat{k}$ and the line of intersection of the planes $\vec{r} \cdot (\hat{i}-2 \hat{j}+3 \hat{k})=5$ and $\vec{r} \cdot (3 \hat{i}+\hat{j}-2 \hat{k})=7$.

• A
  $\vec{r} \cdot (-2 \hat{i}-3 \hat{j}+5 \hat{k})=-2$
• B
  $\vec{r} \cdot (7 \hat{i}-\hat{k})=19$
• C
  $\vec{r} \cdot (4 \hat{i}-\hat{j}+\hat{k})=12$
• D
  $\vec{r} \cdot (8 \hat{i}+5 \hat{j}-9 \hat{k})=16$