Find the equation of the plane passing through the points (-1,1,-4) and is perpendicular to each of the planes
-2x+y+z+2=0;x+y-3z+1=0

Extract the normal vectors from the given planes:

(which are unique up to their signs, meaning either or are valid choices for the normal vector)

The third plane must be perpendicular to both these given planes, which means it would be parallel to both and , which in turn means its own normal vector should be perpendicular to both and .

Enter the cross product:

or (4, 5, 3), which also works.

The given plane passes through (-1, 1, 4), so its equation is

Simplify: