This is a Vandermonde-type determinant. The general Vandermonde determinant for three variables is | 1 a a^2 | | 1 b b^2 | | 1 c c^2 | = (b-a)(c-a)(c-b) = (a-b)(b-c)(c-a). The sign depends on the order of subtraction. Expanding: (a-b)(b-c)(c-a). This is a standard result. The determinant vanishes when any two of a, b, c are equal, which is consistent with the factorization.