Question
Find the angle between the two lines whose direction cosines are given by the equations:
2 l – m + 2 n = 0 and m n + n l + l m = 0
Solution
The given equation are
2 l – m + 2 n = 0 ...(1)
and m n + n l + l m = 0 ....(2)
From (1), m = 2 l + 2 n ....(3)
From (2) and (3), we get,
n (2 l + 2 n) + n l + l (2 l + 2 n) = 0
or 2 n l + 2 n2 + n l + 2 l2 + 2 n l = 0 or 2 l2 + 5 l n + 2 n2 = 0
⇒ (2 l + n) (l + 2 n) = 0
either 2l+ n = 0
i.e. 2l + 0 m + n = 0
Also, 2l - m + 2n = 0
Solving, we get,
or l + 2n = 0
i.e., l + 0 m + 2 n = 0
Also,
2l - m + 2n = 0
Solving, we get,
∴ direction-ratios of two lines are 1, – 2, – 2 and 2, 2, – 1.
Let θ be the angle between the lines
∴ θ = 90°
Let θ be the angle between the lines
∴ θ = 90°
