Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 15 / a / i

The scalar triple product is the determinant of the matrix with columns $a,b,c$. A cyclic permutation of three columns is a product of two swaps, so its determinant sign is positive. Therefore $[a,b,c]=[b,c,a]$. This uses the alternating multilinearity of the determinant, and remains true for linearly dependent vectors.