Whatcan the answer be?

In Part 2 of this series, we saw how dual vectors arose very naturally even in elementary vector analysis. At the end of that article, I stated that dual vectors and the reciprocal basis were very far-reaching concepts. They appear in many different contexts, some of which will be described in the sequel. In this part, we begin with a situation that might appear to be simpler than the case worked out in part 2-but we are in for a surprise!