Find the equation of the plane under the given conditions

Question

Question: Find the equation of the plane passing through the point $(8\hat{i}+2\hat{j}-3\hat{k})$ and perpendicular to each of the planes $\vec{r}\cdot (2\hat{j}-\hat{j}+2\hat{k})=0$ and $\vec{r}\cdot (\hat{i}+3\hat{j}-5\hat{k})+5=0$.

My try: Let $\vec{a}=(8,2,-3)$ and $\hat{n}$ be the unit vector normal to the required plane. Then the equation of the plane is $(\vec{r}-\vec{a})\cdot \hat{n}=0$. What can I do next?

Answer 1

Next to obtain the vector $\vec{n}=\vec{n}_1\times\vec{n}_2$, where $\vec{n}_1= (2,-1,2)$ and $\vec{n}_2 = (1,3,-5)$. Thus, the equation of the plane is

$$(\vec{r}-\vec{a})\cdot (\vec{n}_1\times \vec{n}_2 )=0$$

Answer 2

Hint:

The normal vector of the required plane is perpendicular to the normal vectors of the given two planes.