This formula is identical in structure to the determinant formula for the normal cross product in R3 except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v1, ..., v"n"−1, Λv"i") have a positive orientation with respect to (e1, ..., e"n"). If "n" is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that "n" is even, however, the distinction must be kept. This formula_135-ary form enjoys many of the same properties as the vector cross product: it is alternating and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments.