Find a vector parallel to the line of intersection of the planes given by 2x + z = 5 and x + y - z = 4

The normal to the given plane P1: 2x+z=5 is N1=, and similarly

the normal to the given plane P2: x+y-z=4 is N2=, 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

=V

A vector parallel to the intersection of P1 and P2 is V.

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.