Question

Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l² + m² – n² = 0.

Answer 1

Given equations are,

l + m + n = 0 ….. (i)

l² + m² – n² = 0 …. (ii)

From equation (i), we have n = – (l + m)

Putting the value of n is equation (ii), we get

l² + m² + [-(l + m)]² = 0

l² + m² – l² – m² – 2lm = 0

-2lm = 0

lm = 0 ⇒ (- m – n)m = 0 [Since, l = – m – n]

(m + n)m = 0 ⇒ m = 0 or m = -n

⇒ l = 0 or l = -n

Now, the direction cosines of the two lines are

0, -n, n and -n, 0, n ⇒ 0, -1, 1 and -1, 0, 1

Thus, the required angle π/3.