The angle between the lines whose direction cosines satisfy the equations l ⁡ + m + n = 0 and l &#8289…

The angle between the lines whose direction cosines satisfy the equations l⁡+m+n=0 and l⁡2=m2+n2 is 
  1. π 6
  2. π 2
  3. π 3
  4. π 4

Solution

l⁡+m+n=0

m+n=-l⁡

∴   m+n2=l⁡2

∴   l⁡2= m2+n2+2mn

But, m2+n2=l⁡2

∴   2mn=0

∴  m = 0 or n = 0

∴  l⁡=-n or l⁡=-m

∵l⁡2+m2+n2=1

∴  The two direction cosines are 12,0,-12 and 12,-12,0

θ  is the angle between them.

∴   cos θ=l1l2+m1m2+n1n2=1212+0+0=12

∴    θ = π 3 .

Note : Taking l⁡ = - 1 2 , will also given the same solution, but the supplementary angle. Both are correct, we choose the one present in the options.

Asked in: JEE Main 2014 (06 Apr)

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