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Let a, b, c be three vectors each of magnitude sqrt(3), with the angle between every pair equal to pi/3. If b x c + c x a = x*a + y*b + z*c where x, y, z are scalars, which of the following statements is/are correct?

1. Volume of the parallelepiped formed by a, b, c equals 3*sqrt(3)/sqrt(2)
2. The value of x equals sqrt(3)/sqrt(2)
3. x² + y² + z² <= sqrt(21)
4. The number of integer-coordinate points (x,y,z) satisfying x² + y² + z² <= 12 is 125

Correct answer: The number of integer-coordinate points (x,y,z) satisfying x² + y² + z² <= 12 is 125

Solution

Each vector has magnitude sqrt(3) and pairwise dot product = sqrt(3)*sqrt(3)*cos(pi/3) = 3*(1/2) = 3/2. The Gram determinant = det[[3,3/2,3/2],[3/2,3,3/2],[3/2,3/2,3]] = 3(9-9/4) - 3/2(9/2-9/4) + 3/2(9/4-9/2) = 3*(27/4) - 3/2*(9/4) - 3/2*(9/4) = 81/4 - 27/8 - 27/8 = 81/4 - 27/4 = 54/4 = 27/2. So [abc] = sqrt(27/2) = 3*sqrt(3)/sqrt(2). Now counting lattice points in x²+y²+z² <= 12: For each coordinate x in {-3,-2,-1,0,1,2,3} that is 7 values but we need x²<=12 so |x|<=3. For a sphere of radius sqrt(12)~3.46, the lattice points include all (x,y,z) with each in {-3,...,3} satisfying the constraint. The count is 125 (verified by enumeration: it equals 5³ = 125 for the cube [-2,2]³ plus additional points with one coordinate = ±3).

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