The normal to the given plane P1: 2x+z=5 is N1=<2,0,1>, and similarly
the normal to the given plane P2: x+y-z=4 is N2=<1,1,-1>, and similarly
The normal to the above normals is parallel to the intersection of planes P1 and P2, which is given by the cross product of N1 and N2:
V = N1 x N2 =
i j k
2 0 1
1 1 -1
= V< 0-1, 1-(-2), 2-0>
=V<-1,3,2>
A vector parallel to the intersection of P1 and P2 is V<-1,3,2>.
Check:
substitute V in P1: 2(-1)+0+(2)=0 => V is parallel to P1.
substitute V in P2: (-1)+(3)-(2)=0 => V is parallel to P2
Therefore V is parallel to intersection of P1 and P2.
