Chapter 11 Three Dimensional Geometry Worksheet Page 29
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41THREE DIMENSIONAL GEOMETRY
491
which is the length of the perpendicular from a point to the given plane.
We may establish the similar results for (Fig 11.19 (b)).
A
Note
KH
KH
H
If the equation of the plane π
⋅
=
1.
is in the form
r
N
d
, where
is normal
N
2
KH
H
⋅
−
|
a
N
d
|
KH
.
to the plane, then the perpendicular distance is
| N |
KH
H
|
d
|
KH
⋅
=
2.
The length of the perpendicular from origin O to the plane
is
r
N
d
| N |
H
(since
a
= 0).
Cartesian form
H
Let P(x
, y
, z
) be the given point with position vector
a
and
1
1
1
Ax + By + Cz = D
be the Cartesian equation of the given plane. Then
H
+
+
ˆ
ˆ
ˆ
x i
y j
z k
a
=
1
1
1
KH
+
+
ˆ
ˆ
ˆ
A
i
B
j
C
k
=
N
Hence, from Note 1, the perpendicular from P to the plane is
+
+
⋅
+
+
−
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
x i
y j
z k
) ( A
i
B
j
C
k
)
D
1
1
1
+
+
2
2
2
A
B
C
+
+
−
A
x
B
y
C
z
D
1
1
1
=
+
+
2
2
2
A
B
C
Example 24
Find the distance of a point (2, 5, – 3) from the plane
H
⋅
−
+
ˆ
ˆ
ˆ
= 4
r
( 6
i
3
j
2
k
)
KH
H
=
+
−
=
−
+
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Solution
Here,
and d = 4.
a
2
i
5
j
3 , N
k
6
i
3
j
2
k
Therefore, the distance of the point (2, 5, – 3) from the given plane is
+
−
⋅
−
+
−
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
−
−
−
| (2
i
5
j
3 ) (6
k
i
3
j
2 )
k
4|
| 12 15
6
4 |
13
=
=
−
+
ˆ
ˆ
ˆ
+ +
| 6
i
3
j
2 |
k
7
36
9 4
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