Find the vector equation of the plane passing through the intersection of the planes r ·(2 i +2 j -3 k )=7 and r ·(2 i +5 j +3 k )=9 and through the…

Equation of the plane is \(\displaystyle \vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})-7+\lambda[\vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})-9]=0\) \[\begin{aligned} & \text { or } \vec{r} \cdot[(2+2 \lambda) \hat{i}+(2+5 \lambda) \hat{j}+(-3+3 \lambda) \hat{k}]-7-9 \lambda \\ & \quad(2,1,3) \text { lies on it } \\ & \therefore 2(2+2 \lambda)+1(2+5 \lambda)+3(-3+3 \lambda)-7-9 \lambda=0 \\ & \quad 9 \lambda=10 \Rightarrow \lambda=10 / 9 \end{aligned} \] Equation is \(\displaystyle \vec{r} \cdot(38 \hat{i}+68 \hat{j}+3 \hat{k})=153\)