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Three non-coplanar unit vectors a, b, and c are pairwise inclined at the same acute angle θ, and they are not collinear. The value of |a |b |c| in terms of θ is

1. (1+cosθ)√(cos 2θ)
2. (1+cosθ)√(1-2cos 2θ)
3. (1-cosθ)√(1+2cosθ)
4. None of these

Correct answer: (1-cosθ)√(1+2cosθ)

Solution

|[a b c]|^2 equals the Gram determinant with 1 on the diagonal and cos(theta) off-diagonal = 1 - 3cos^2(theta) + 2cos^3(theta) = (1-cos theta)^2(1+2cos theta). So |[a b c]| = (1-cos theta)*sqrt(1+2cos theta).

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