Scalar Triple Product

The scalar triple product [a b c] equals a . (b x c) and is evaluated as the determinant whose rows are the components of the three vectors. Geometrically it gives the signed volume of the parallelepiped spanned by the vectors, and its vanishing signals coplanarity, a frequently tested JEE Advanced criterion. Setting up the determinant with rows (1,2,3), (2,1,1), (3,1,2), expand along the first row: 1[(1)(2) - (1)(1)] - 2[(2)(2) - (1)(3)] + 3[(2)(1) - (1)(3)] = 1(2 - 1) - 2(4 - 3) + 3(2 - 3) = 1(1) - 2(1) + 3(-1) = 1 - 2 - 3 = -4. Option 6 takes the wrong sign and an inflated magnitude; option 0 would mean the vectors are coplanar, which the nonzero determinant refutes; option -12 triples a minor erroneously. Plausibility check: since the determinant is nonzero, the three vectors are linearly independent and span a genuine three-dimensional parallelepiped, consistent with a nonzero volume.