Chapter 11 Three Dimensional Geometry Worksheet Page 13
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41THREE DIMENSIONAL GEOMETRY
475
Hence, the required shortest distance is
d = PQ = ST | cos θ |
H
H
H
H
×
−
(
b
b
) . (
a
a
)
H
H
1
2
2
1
or
d =
×
|
b
b
|
1
2
Cartesian form
The shortest distance between the lines
−
−
−
x
x
y
y
z
z
=
1
1
1
l
:
=
1
a
b
c
1
1
1
−
−
−
x
x
y
y
z
z
=
2
2
2
and
l
:
=
2
a
b
c
2
2
2
−
−
−
x
x
y
y
z
z
2
1
2
1
2
1
a
b
c
1
1
1
is
a
b
c
2
2
2
−
+
−
+
−
2
2
2
b c
b c
c a
c a
a b
a b
(
)
(
)
(
)
1 2
2 1
1 2
2 1
1 2
2 1
11.5.2 Distance between parallel lines
If two lines l
and l
are parallel, then they are coplanar. Let the lines be given by
1
2
H
H
H
+ λ
=
... (1)
a
b
r
1
H
H
H
+ µ
and
=
… (2)
a
b
r
2
H
where,
is the position vector of a point S on l
and
a
1
1
H
a
is the position vector of a point T on l
Fig 11.9.
2
2
As l
, l
are coplanar, if the foot of the perpendicular
1
2
from T on the line l
is P, then the distance between the
1
lines l
and l
= | TP |.
1
2
H
KKH
Let θ be the angle between the vectors
and
b
.
ST
Fig 11.9
Then
H
KKH
H
KKH
b ×
θ
=
ˆ
... (3)
ST
( | | | ST| sin )
b
n
where
n ˆ
is the unit vector perpendicular to the plane of the lines l
and l
1
2.
KKH
H
H
−
a
a
But
=
ST
2
1
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