A plane satisfies the dot product equation equation
where r is the position vector [x y z] and v = [vx vy vz] is a vector perpendicular to the plane. A vector perpendicular to the first plane is
because dotting that into [x y z] produces the equation for the first plane. Similarly for the other two. A unit vector perpendicular to the plane is
and similarly for the other two. Then dot(n1,n2) tellls you if n1 and n2 are orthogonal, and similarly for the other possible pairs.
Planes are orthogonal if and only if their normal vectors are orthogonal.