A couple of comments.
1. You're confusing your notation there. You've got for the ith vector, but also for the ith component of one of those vectors. Better to write for the jth component of the ith vector, and sum over j in the norm.
2. I'm a bit surprised at the result. I would have expected the order to reverse:
That last step is valid because everything in sight is a unit vector. So, in comparing with the norm of the difference, you've got a sign reversal. Hence, I would expect the order to reverse.