� Cartesian equation of a line that passes through two points (x1, y1, z1) and
 (x2, y2, z2) is 
1
1
1
2
1
2
1
2
1
x
x
y
y
z
z
x
x
y
y
z
z
−
−
−
=
=
−
−
− � If θ is the acute angle between 
1
1
r
a
b
=
+ λ
r
r
r
 and 
2
2
r
a
b
=
+ λ
r
r
r
, then
1
2
1
2
cos
|
| |
|
bb b
b
⋅
θ =
r
r
r
r
� If  
1
1
1
1
1
1
n
z
z
m
y
y
l
x
x
−
=
−
=
−
 and  
2
2
2
2
2
2
n
z
z
m
y
y
l
x
x
−
=
−
=
−
are the equations of two lines,  then the  acute angle between the two lines is
given by cos θ  = |l1l2 + m 1m 2 + n1n2| © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
501
� Shortest distance between two skew lines is the line segment perpendicular
to both the lines � 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