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Question
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0.
Solution
The given equations are
l + m + n = 0 ......(i)
l2 + m2 – n2 = 0 .......(ii)
From equation (i) n = – (l + m)
Putting the value of n in equation (ii) we get
l2 + m2 + [– (l + m)2] = 0
⇒ l2 + m2 – l2 – m2 – 2lm = 0
⇒ – 2lm = 0
⇒ lm = 0
⇒ (– m – n)m = 0 .....[∵ l = – m – n]
⇒ (m + n)m = 0
⇒ m = 0 or m = – n
⇒ l = 0 or l = – n
∴ Direction cosines of the two lines are
0, – n, n and – n, 0, n
⇒ 0, – 1, 1 and – 1, 0, 1
∴ `cos theta = ((0hat"i" - hat"j" + hat"k")*(-hat"i" + 0hat"j" + hat"k"))/(sqrt((-1)^2 + (1)^2) sqrt((-1)^2 + (1)^2)`
= `1/(sqrt(2) * sqrt(2)`
= `1/2`
∴ `theta = pi/3`
Hence, the required angle is `pi/3`.
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