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NDA 2019 Maths · Mathematics · Three Dimensional Geometry · hard

The sine of the angle between the line with direction ratios (1,2,2) and the plane 2x-y+2z=5 is

Single correct option

The sine of the angle between the line with direction ratios (1,2,2)(1,2,2) and the plane 2x−y+2z=52x-y+2z=5 is
  1. A. 59\frac59
  2. B. 43\frac{4}{3}
  3. C. 659\frac{\sqrt{65}}{9}
  4. D. 49\frac49
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Answer: (D) 49\frac49

The angle ϕ\phi between a line with direction vector d⃗\vec d and a plane with normal n⃗\vec n satisfies sin⁡ϕ=∣d⃗⋅n⃗∣∣d⃗∣∣n⃗∣\sin\phi=\frac{|\vec d\cdot\vec n|}{|\vec d||\vec n|}. Here d⃗⋅n⃗=2−2+4=4\vec d\cdot\vec n=2-2+4=4 and ∣d⃗∣∣n⃗∣=3⋅3=9|\vec d||\vec n|=3\cdot3=9, so sin⁡ϕ=49\sin\phi=\frac49. Using cos⁡\cos in place of sin⁡\sin gives 659\frac{\sqrt{65}}{9}, the usual trap.
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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