[gbjw]

To be very explicit, if |x| = |y| = 1, we have |x - y|^2 = |x|^2 - 2xy + |y|^2 = 2 - 2xy = 2 - 2* cos(th). So they are not identical but minimizing the Euclidian distance of two unit vectors is the same as maximizing the cosine similarity.