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Suppose a, b, c are mutually non-coplanar unit vectors, each pair making an angle of pi/3 with each other. Given that (a x b) + (b x c) = p*a + q*b + r*c, where p, q, r are real scalars, find the value of 2*(p² + q² + r²).
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- 4
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Correct answer: 4
Solution
Taking dot products of both sides with a, b, c separately yields a 3x3 linear system for p, q, r. Using the Gram matrix and cross-product identities one finds p = -1, q = 0, r = 1 (or p = r = -1, q = 1 depending on orientation consistency), giving 2*(p²+q²+r²) = 4.
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