We don't know the length, so we cannot use the vectors by themselves., but there are sequences that are very similar to them that will have all the properties. You can assume it is an infinite sequences.

4. Let R be a unital ring. Show that the ring S = S(R) has elements e1, ez, e3 satisfying e? = ei (i = 1, 2, 3) eiej = 0s (i* j) e1 + e2 + e3 = 1s Hint: Think about the standard basis vectors e1, e2, e3 in Rs, which sum to the vector (1, 1, 1). Can you use this idea to find three sequences in S that sum to the identity sequence