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1
Distances Between Points, Planes,
and Lines
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This section is concluded with the following discussion of
two basic types of problems involving distance in space.
1. Finding the distance between a point and a plane
2. Finding the distance between a point and a line
The distance
D
between a point
Q
and
a plane is the length of the shortest line
segment connecting
Q
to the plane,
as shown in Figure 11.52.
Figure 11.52
Distances Between Points, Planes, and Lines
3
Distances Between Points, Planes, and Lines
If
P
is
any
point in the plane, you can find this distance by
projecting the vector
onto the normal vector
n
. The
length of this projection is the desired distance.
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Example 5 –
Finding the Distance Between a Point and a Plane
Find the distance between the point
Q
(1, 5, –4 ) and the
plane given by 3
x
–
y
+ 2
z
= 6.
Solution:
You know that
is normal to the given plane.
To find a point in the plane, let
y
= 0 and
z
= 0 and obtain
the point
P
(2, 0, 0).
The vector from
P
to
Q
is given by
5
Example 5 –
Solutions
Using the Distance Formula given in Theorem 11.13
produces
cont’d
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This note was uploaded on 10/05/2011 for the course BIO 203, CH taught by Professor Lacey,simmerling,deng,hanson during the Fall '10 term at SUNY Stony Brook.
 Fall '10
 Lacey,Simmerling,Deng,Hanson

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