multiplying the 1st by -2 and adding it to the 2nd gives
Now we can parameterize the line. Let then and finally x is
so we get
We can now find the normal vector from the plane to be
Now if we cross these two vectors we will get a vector that is perpendicular to both of them so it is parallel to the plane.
From our work before we know the point is in the plane and let (x,y,z) be any other point in the plane then the vector
is in the plane.
Now If we dot v with the vector from the cross product and set it equal to zero we will get the equation of the plane perpendicular to the original plane.