We have
2l + 2m - n = 0 ...(1)
mn + nl + lm = 0 ...(2)
From Eq. (1), we have n = 2l + 2m. Substituting the value of n in Eq. (2), we have
m(2l + 2m) + l(2l + 2m) + lm = 0
2l2 + 5lm + 2m2 = 0
(2l + m)(l + 2m) = 0
m = -2l, l = -2m
Now m = -2l and n = 2l + 2m = 2l - 4l = -2l. This implies that
l/1 = m/-2 = n/-2
l/-1 = m/2 = n/2
Hence l = -2m and n = 2l + 2m = - 2m. This implies that
l/2 = m/-1 = n/2
Therefore, the DRs of the lines are (−1, 2, 2) and (2, −1, 2) and the dot product is given by −2 − 2 + 4 = 0. Hence, the lines are at right angles to each other.
