If PQ
uuur
 is the shortest distance vector between
l1 and l2 , then it being perpendicular to both  1b
r
 and
2b
r
, the unit vector nˆ  along PQ
uuur
 would therefore be
ˆn =
1
2
1
2
|
|
b
b
b
b
×
×
r
r
r
r (3)
Then
PQ
uuur
 = d nˆ
where, d is the magnitude of the shortest distance vector Let θ be the angle between
ST
uur
 and PQ
uuur Then
PQ = ST |cos θ|
But
cos θ =
PQ ST
| PQ | | ST |
⋅
uuur uur
uuuur
uur
 =
2
1
ˆ (
)
ST
d n
a
a
⋅d
−
r
r
(since  
2
1
ST
)
a
a
=
−
uur
r
r
=
1
2
2
1
1
2
(
) (
)
ST
b
b
a
a
b
b
×
⋅
−
×
r
r
r
r
r
r
[From (3)]
Fig 11