If the direction cosines (l,m,n) of two lines are connected by the relations l+m+n=0 and lm=0, then the angle between those lines is

cosθ=−→b1.→b2−→|b1|∣∣∣→b2∣∣∣ ...... (1)

l+m+n=0

l+m=−n

−(l+m)=n

And, lm=0

So,

Either l=0 or m=0

When, l=0, then

m=−n

And,

(l,m,n)=(0,1,−1)

When, m=0, then

l=−n

And,

(I,m,n)=(1,0,−1)

Calculate b1⋅b2.

b1.b2=(0,1,−1).(1,0,−1)

=0+0+1

=1

|b1|=√02+12+(−1)2=√2

|b2|=√02+12+(−1)2=√2

cosθ=1√2.√2=1/2

θ=π/3

Hence, the angle between the lines is π/3.