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AP EAMCET2013MathematicsThree Dimensional Geometry

The shortest distance between the lines r =3 i +5 j +7 k + ( i +2 j + k ) and r =- i - j - k + (7 i -6 j + k ) is

Options

  1. A16 5 5
  2. B26 5 5
  3. C36 5 5
  4. D46 5 5

Correct answer

D. 46 5 5

Step-by-step solution

The given lines are r = a ₁+ b ₁, r = a ₂+ b ₂ where, aligned & a ₁=3 i +5 j +7 k , b ₁= i +2 j + k & a ₂=- i - j - k , b ₂=7 i -6 j + k & | b ₁ b ₂ |= | array ccc i & j & k 1 & 2 & 1 7 & -6 & 1 array | & | i (2+6)- j (1-7)+ k (-6-14)| & |8 i +6 j -20 k | & 64+36+400 = 500 =10 5 aligned Now, aligned & [ ( a ₂- a ₁ ) b ₁ b ₂ ]= ( a ₂- a ₁ ) ( b ₁ b ₂ ) & =(-4 i -6 j -8 k ) (8 i +6 j -20 k ) & =-32-36+160 & =160-68=92 aligned Shortest distance aligned & = [ ( a ₂- a ₁ ) ( b ₁ b ₂ ) ] | b ₁ b ₂ | & = 92 10 5 = 46 5 5

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