Let v_0 = [z, 0, 0, -w], v_1 = [0, z, 0, -x], v_2 = [0, 0, z, -y]
Let n_0, n_1, n_2 be any random variables
n_0v_0 + n_1v_1 + n_2v_2
= [n_0z, 0, 0, -n_0w] + [0, n_1z, 0, -n_1x] + [0, 0, n_2z, -n_2y]
= [n_0z, n_1z, n_2z, -n_0w - n_1x - n_2y]
(w, x, y, z) . (n_0z, n_1z, n_2z, -n_0w - n_1x - n_2y)
= n_0zw + n_1zx + n_2zy - n_0zw - n_1zx - n_2zy
= 0
as required