� The angle φ between the line r
a
b
=
+ λ
r
r
r
and the plane 
r nˆ
d
⋅
=
r
is
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
503
ˆ
sin
| | | |ˆ
b n
b n
⋅
φ =
r
r
� The angle θ between the planes A1x + B1y + C1z + D1 = 0 and
A2 x + B2 y + C2 z + D2 = 0 is given by
cos θ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
A A
B B
C C
A
B
C
A
B
C
+
+
+
+
+
+
� The distance of a point whose position vector is ar from the plane 
r nˆ
d
⋅
=
r
 is
ˆ
|
|
d
−a n
⋅
r
� The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is
1
1
1
2
2
2
A
B
C
D
A
B
C
x
y
z
+
+
+
+
+ —�—
© NCERT
not to be republished
504
MATHEMATICS
�The mathematical experience of the student is incomplete if he never had
the opportunity to solve a problem invented by himself – G POLYA �
12