if θ = 90° by (1)
a1a2 + b1b2 + c1c2 = 0
(ii)
parallel i e if θ = 0 by (2)
1
2
a
a  =
1
1
2
2
b
c
b
c
=
Now, we find the angle between two lines when their equations are given If θ is
acute the angle between the lines
rr  =
1
1
a
b
+λ
ur
 and  rr  = 
2
2
a
b
+µ
r
r
then
cosθ =
1
2
1
2
b
b
b
b
⋅
r
r
r
r
In Cartesian form, if θ is the angle between the lines
1
1
x
x
a
−
 =
1
1
1
1
y
y
z
z
b
c
−
−
=