[math-fun] Puzzle: Find limiting angle as dimension -> oo

Let v_n be the vector in R^(n-1) defined as (1/n, 2/n,...,(n-1)/n). Let w_n be the vector in R^(n-1) with the same coordinates in reverse order. Find the limit as n -> oo of the angle between v_n and w_n. (Try this without using numerical approximation or summation-of-powers formulae.) Is there a simple geometric reason for the answer? I don't know. --Dan