� If  l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and  θ  is the
acute angle between the two lines; then
cosθ  =  |l1l2 + m 1m 2 + n1n2|
� If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the
acute angle between the two lines; then
cosθ = 
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
a a
b b
c c
a
b
c
a
b
c
+
+
+
+
+
+
� Vector equation of a line that passes through the given point whose position
vector is ar  and parallel to a given vector b
r
 is r
a
b
=
+ λ
r
r
r � Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is
1
1
1
x
x
y
y
z
z
l
m
n
−
−
−
=
=
� The vector equation of a line which passes through two points whose position
vectors are ar  and b
r
 is 
(
)
r
a
b
a
=
+ λ
−
r
r
r
r � 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|