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NDA 2025 Maths · Mathematics · Three Dimensional Geometry · medium

The angle θ between the planes x+y+z=1 and x-y+z=2 satisfies

Single correct option

The angle θ\theta between the planes x+y+z=1x+y+z=1 and x−y+z=2x-y+z=2 satisfies
  1. A. cos⁡θ=13⋅3\cos\theta=\dfrac13\cdot\sqrt3
  2. B. cos⁡θ=0\cos\theta=0
  3. C. cos⁡θ=13\cos\theta=\dfrac13
  4. D. cos⁡θ=23\cos\theta=\dfrac23
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Answer: (C) cos⁡θ=13\cos\theta=\dfrac13

The angle between planes equals the angle between their normals (1,1,1)(1,1,1) and (1,−1,1)(1,-1,1): cos⁡θ=1−1+133=13\cos\theta=\frac{1-1+1}{\sqrt3\sqrt3}=\frac13. The option cos⁡θ=0\cos\theta=0 would be perpendicular planes, which needs a zero dot product, and 23\frac23 arises from adding the absolute values 1+1+1−11+1+1-1.
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