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When we considered lines in R 2 , we showed how to determine whether lines were
parallel or perpendicular. It is possible to use the same formula to determine
whether planes are parallel or perpendicular.
Parallel and Perpendicular Planes
Perpendicular Planes
Parallel Planes
n 1
n 2
p 1
n 2
p 2
n 1
p 1
p 2
p 1
!
1. If and p 2 are two perpendicular planes, with normals and n respectively,
their normals are perpendicular 1that is, n 1!
#
! 2 ,
n2 02.
! !
2. If p 1 and p 2 are two parallel planes, with normals n1 and n respectively,
! ! 2 ,
their normals are parallel 1that is, n 1 kn 2 2 for all nonzero real numbers k.
n1
!
EXAMPLE 4
Reasoning about parallel and perpendicular planes
a. Show that the planes p 1 : 2x 3y z 1 0 and p 2 : 4x 3y 17z 0 are
perpendicular.
b. Show that the planes p 3 : 2x 3y 2z 1 0 and p 4 : 2x 3y 2z 3 0
are parallel but not coincident.
Solution
! !
a. For n1 12, 3, 12 and for p 2 : n 2 14, 3, 172.
p1 :
n 1!
# n2
! 12, 3, 12 # 14, 3, 172
2142 3132 11172
8 9 17
0
Since n 1!
#
!
n2 0, the two planes are perpendicular to each other.
! !
b. For p 3 and p4 , n 3 n4 12, 3, 22, so the planes are parallel. Because the
planes have different constants (that is, 1 and 32, the planes are not coincident.
466 8.5 THE CARTESIAN EQUATION OF A PLANE
NEL