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JEE Main2014MathematicsThree Dimensional GeometryActual

The angle between the lines whose direction cosines satisfy the equations l ⁡ + m + n = 0 and l ⁡ 2 = m 2 + n 2 is

Options

  1. Aπ 6
  2. Bπ 2
  3. Cπ 3
  4. Dπ 4

Correct answer

C. π 3

Step-by-step solution

l ⁡+ m + n =0 m + n =- l ⁡ ∴     m + n 2 = l ⁡ 2 ∴    l ⁡ 2 =   m 2 + n 2 + 2 m n But, m 2 + n 2 = l ⁡ 2 ∴    2 m n =0 ∴ m   =   0 or n   =   0 ∴ l ⁡ = - n or l ⁡ = - m ∵ l ⁡ 2 + m 2 + n 2 = 1 ∴ The two direction cosines are 1 2 , 0 , - 1 2 and 1 2 , - 1 2 , 0 θ is the angle between them. ∴    cos  θ = l 1 l 2 + m 1 m 2 +

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