A distance is a function d(x, y) where d(x, y) = 0 if x = y, if x = 0, or if y = 0 (among other properties). If x = y and neither x = 0 nor y = 0, and if we let

A distance is a function d(x, y) where d(x, y) = 0 if x = y, if x = 0, or if y = 0 (among other properties). If x = y and neither x = 0 nor y = 0, and if we let d(x, y) = x*y, then d(x, y) != 0 in general, so the dot product is certainly not a distance. Your comment isn't relevant, but thanks for calling me a pedant. ;-)