If the planes are parallel, then 
1
1
1
2
2
2
A
B
C
A
B
= =C Example 22 Find the angle between the two planes 2x + y – 2z = 5 and 3x – 6y – 2z = 7
using vector method Solution The angle between two planes is the angle between their normals From the
equation of the planes, the normal vectors are
N1
ur
 =
ˆ
ˆ
ˆ
2
2
i
j
k
+
−
 and  
2
ˆ
ˆ
ˆ
N
3
6
2
i
j
k
=
−
−
ur
Therefore
cos θ =
1
2
1
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
N
N
(2
2
) (3
6
2
)
| N | |N |
4
1
4
9
36
4
i
j
k
i
j
k
⋅
+
−
⋅
−
−
=
+
+
+
+
ur
ur
ur
ur
 = 
4
21






Hence
θ = cos – 1


21
4
© NCERT
not to be republished
 MATHEMATI CS
490
Example 23 Find the angle between the two planes 3x – 6y + 2z = 7 and 2x + 2y – 2z =5