Equation of a $ 3$rd plane - two points and parallel to the line of intersection of two planes

Question

I have to find the Equation of a plane $A$

Data I do have : Two points on that plane $A$, Equations of two planes, whose line of intersection is parallel to plane $A$

1. I found a vector $v_1$ with the two points on the plane $A$ (So that vector lies on the plane $A$).

2. From two given planes, I calculated their cross product. It gives the vector $v_2$, which is the line of intersection.

3. Now I know the vector $v_1$ is parallel to the $v_2$. So, the normal of vector $v_1$ is parallel to normal of vector $v_2$.

With all these information how can I find the equation of plane $A$ ?

Answer 1

If the two planes have normals $\vec n_1$ and $\vec n_2$, then $\vec n_1 \times \vec n_2$ will give a vector that is parallel to the line of intersection of the planes and thus parallel to the plane of interest. Call $\vec n_1 \times \vec n_2=\vec v$. You have two points so you can find another vector that lies in the plane call this $\vec w$. Taking the cross product of $\vec w$ and $\vec v$ will give you a normal to the plane of interest.

This assumes of course that your two points adds information about the plane. In other words they do not all lie on the line. Otherwise, there are an infinite amount of planes that fit the criteria.

Answer 2

Hint: use the fact that since the new plane passes through the intersection of both the planes , it will satisfy the equations of both the planes. Therefore Plane 1 + (c)*Pane 2 =0 where C is a constant. Use the given point and the direction ratioS