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NDA 2020 Maths · Mathematics · Three Dimensional Geometry · medium

The cosine of the angle between two lines with direction ratios (1,2,2) and (2,-3,6) is

Single correct option

The cosine of the angle between two lines with direction ratios (1,2,2)(1,2,2) and (2,−3,6)(2,-3,6) is
  1. A. 863\dfrac{8}{63}
  2. B. 821\dfrac{8}{21}
  3. C. 47\dfrac{4}{7}
  4. D. 37\dfrac{3}{7}
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Answer: (B) 821\dfrac{8}{21}

The dot product is 1⋅2+2⋅(−3)+2⋅6=81\cdot2+2\cdot(-3)+2\cdot6=8, and the magnitudes are 1+4+4=3\sqrt{1+4+4}=3 and 4+9+36=7\sqrt{4+9+36}=7. Hence cos⁡θ=83⋅7=821\cos\theta=\dfrac{8}{3\cdot7}=\dfrac8{21}. The value 863\dfrac8{63} multiplies an extra factor 33 into the denominator, and 37\dfrac37 is simply the ratio of the magnitudes. Both direction ratio sets need normalising, not just one.
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Original question written in the NDA 2020 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