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JEE Main 2017 · Mathematics · Three Dimensional Geometry · medium

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

Single correct option

The cosine of the angle between the lines with direction ratios (1,2,2)(1,2,2) and (2,3,6)(2,3,6) is
  1. A. 2021\dfrac{20}{21}
  2. B. 1021\dfrac{10}{21}
  3. C. 47\dfrac{4}{7}
  4. D. 23\dfrac{2}{3}
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Answer: (A) 2021\dfrac{20}{21}

Use cos⁡θ=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22\cos\theta=\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}. The numerator is 2+6+12=202+6+12=20. The magnitudes are 1+4+4=3\sqrt{1+4+4}=3 and 4+9+36=7\sqrt{4+9+36}=7. So cos⁡θ=2021\cos\theta=\dfrac{20}{21}. Note that the lines are nearly parallel, which fits the large cosine value.
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