Find the angle between the line whose direction cosines are given by `l+m+n=0a n d2l^2+2m^2-n^2=0.`

Explanation

`l^(2)+m^(2)+n^(2)=1" "`(i)
`" "l+m+n=0" "`(ii)
`" "2l^(2)+2m^(2) -n^(2)=0" "`(iii)
`" "2(l-n^(2))=n^(2)or 3n^(2)=2 or n=pmsqrt(2//3)" "`(iv)
`" "2(l^(2)+m^(2))=n^(2)=(-(l+m))^(2)or l=m" "`(v)
`" "l+m=pmsqrt(2//3) or 2l=pmsqrt(2//3)`
`" "l=pm1//sqrt(6), m=pm1//sqrt(6)`
Direction cosines are
`" "((1)/(sqrt(6)),(1)/(sqrt(6)), sqrt((2)/(3)))and ((1)/(sqrt(6)),(1)/(sqrt(6)),-sqrt((2)/(3)))`
or
`" "(-(1)/(sqrt(6)),-(1)/(sqrt(6)), sqrt((2)/(3)))and (-(1)/(sqrt(6)), -(1)/(sqrt(6)), -sqrt((2)/(3)))`
The angle between these lines in both the cases is `cos^(-1)(-(1)/(3))`.