Cartesian form
Let P(x1, y1, z1) be the given point with position vector ar and
Ax + By + Cz = D
be the Cartesian equation of the given plane Then
ar  =
1
1
1 ˆ
ˆ
ˆ
x i
y
j
z k
+
+
N
ur
 =
ˆ
ˆ
ˆ
A
B
C
i
j
k
+
+
Hence, from Note 1, the perpendicular from P to the plane is
1
1
1
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
) ( A
B
C
)
D
A
B
C
x i
y
j
z k
i
j
k
+
+
⋅
+
+
−
+
+
=
1
1
1
2
2
2
A
B
C
D
A
B
C
x
y
z
+
+
−
+ +
Example 24 Find the distance of a point (2, 5, – 3) from the plane
ˆ
ˆ
ˆ
( 6
3
2
)
r
i
j
k
⋅
−
+
r
 = 4
Solution Here,  
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
5
3
, N
6
3
2
= + −
= − +
ur
ra
i
j
k
i
j
k and d = 4 Therefore, the distance of the point (2, 5, – 3) from the given plane is
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
| (2
5
3 ) (6
3
2
)
4|
ˆ
ˆ
ˆ
| 6
3
2
|
i
j
k
i
j
k
i
j
k
+
−
⋅
−
+
−
−
+
 = 
| 12
15
6
4 |
713
36
9
4
−
−
−
=
+
+
© NCERT
not to be republished
 MATHEMATI CS
492
11 10  Angle between a Line and a Plane
Definition 3 The angle between a line and a plane is
the complement of the angle between the line and
normal to the plane (Fig 11