� Shortest distance between 
1
1
r
a
b
= +λ
r
r
r
 and 
2
2
r
a
b
= +µ
r
r
r
 is
1
2
2
1
1
2
(
) (
–
)
|
|
b
b
a
a
b
b
×
⋅
×
r
r
r
r
r
r
� Shortest distance between the lines: 
1
1
1
1
1
1
x
x
y
y
z
z
a
b
c
−
−
−
=
=
 and
2
2
2
2
x
x
y
y
a
b
−
−
=
 = 
2
2
z
z
−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
bc
b c
c a
c a
a b
a b
−
−
−
−
+
−
+
−
� Distance between parallel lines 
1
r
a
b
= +λ
r
r
r
and 
2
r
a
b
= +µ
r
r
r
is
2
1
(
)
|
|
b
a
a
b
×
−
r
r
r
r
� In the vector form, equation of a plane which is at a distance d from the
origin, and nˆ  is the unit vector normal to the plane through the origin is
r nˆ
d
⋅
=
r � Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d � The equation of a plane through a point whose position vector is ar  and
perpendicular to the vector N
ur
 is (
) N
0
r
−a
=
ur
r
r