Question
Find the angle between the two lines whose direction cosines are given by the equations:
l + m + n = 0 and 2 l + 2 m – m n = 0
Solution
The given equations are
l + m + n = 0 ....(1)
and 2 l + 2 m – m n = 0 ...(2)
From (1), l = – (m + n) ...(3)
From (2) and (3), we get,
– 2 (m + n) + 2 m – m n = 0 or – 2 n – m n = 0
⇒ n (2 + m) = 0
Either n = 0
i.e., 0 l + 0 m + n = 0
Also,
l + m + n = 0
Solving, we get,
or 2 + m = 0 i.e., m = -2
from (1), l - 2 + n = 0
Now, l = 1, n = 1 satisfy it
Also, l = 1, m =-2, n = 1 satisfy (2)
we have
l = 1, m = -2, n = 1
∴ direction-ratios of two lines are 1, – 1, 0 and 1, –2, 1.
Let θ be the angle between the lines
