8
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
475
Hence, the required shortest distance is
d = PQ = ST |cos θ|
or
d =
1
2
2
1
1
2
(
) (
)
|
|
b
b
a
a
b
b
×
−
×
r
r
r
r
r
r
Cartesian form
The shortest distance between the lines
l1 : 
1
1
x
x
−a
 =
1
1
1
1
y
y
z
z
b
c
−
−
=
and
l2  : 
2
2
x
x
−a
 =
2
2
2
2
y
y
z
z
b
c
−
−
=
is
2
1
2
1
2
1
1
1
1
2
2
2
2
2
2
1 2
2 1
1
2
2 1
1 2
2 1
(
)
(
)
(
)
x
x
y
y
z
z
a
b
c
a
b
c
b c
b c
c a
c a
a b
a b
−
−
−
−
+
−
+
−
11 5 2 Distance between parallel lines
If two lines l1 and  l2 are parallel, then they are coplanar