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NDA 2024 Maths · Mathematics · Vector Algebra · medium

The angle which the vector hat i+hat j+hat k makes with the x-axis is

Single correct option

The angle which the vector i^+j^+k^\hat i+\hat j+\hat k makes with the xx-axis is
  1. A. cos⁡−113\cos^{-1}\frac13
  2. B. 45∘45^\circ
  3. C. cos⁡−113\cos^{-1}\frac{1}{\sqrt3}
  4. D. 60∘60^\circ
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Answer: (C) cos⁡−113\cos^{-1}\frac{1}{\sqrt3}

The xx-axis has unit vector i^\hat i, so cos⁡θ=i^⋅(i^+j^+k^)∣i^+j^+k^∣=13\cos\theta=\frac{\hat i\cdot(\hat i+\hat j+\hat k)}{|\hat i+\hat j+\hat k|}=\frac1{\sqrt3}. Hence θ=cos⁡−113≈54.7∘\theta=\cos^{-1}\frac1{\sqrt3}\approx54.7^\circ. Trap: 13\frac13 forgets the square root of the magnitude, and 45∘45^\circ or 60∘60^\circ are guessed from nicer-looking cosines.
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