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NDA 2019 Maths · Mathematics · Three Dimensional Geometry · hard

A line is perpendicular to each of two lines whose direction ratios are (1,2,2) and (2,1,-2). Its direction cosines are

Single correct option

A line is perpendicular to each of two lines whose direction ratios are (1,2,2)(1,2,2) and (2,1,−2)(2,1,-2). Its direction cosines are
  1. A. ±(23,23,−13)\pm\left(\frac23,\frac23,-\frac13\right)
  2. B. ±(23,−23,13)\pm\left(\frac23,-\frac23,\frac13\right)
  3. C. ±(13,23,23)\pm\left(\frac13,\frac23,\frac23\right)
  4. D. ±13(1,1,1)\pm\frac{1}{\sqrt3}(1,1,1)
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Answer: (B) ±(23,−23,13)\pm\left(\frac23,-\frac23,\frac13\right)

A common perpendicular has direction ratios along the cross product (1,2,2)×(2,1,−2)=(−6,6,−3)(1,2,2)\times(2,1,-2)=(-6,6,-3), proportional to (2,−2,1)(2,-2,1). Its magnitude is 33, giving ±(23,−23,13)\pm\left(\frac23,-\frac23,\frac13\right). Check by dot products: (2,−2,1)⋅(1,2,2)=0(2,-2,1)\cdot(1,2,2)=0 and (2,−2,1)⋅(2,1,−2)=0(2,-2,1)\cdot(2,1,-2)=0. Both signs are valid directions.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake