Attempting to find a relation that satisfies an equation (given in text)

I'm trying to find out if there is some sort of functional relationship between and such that . In other words, under what conditions (i.e. relationship between and would this equation be true?

With some manipulating, , and also , but I'm having trouble figuring out how to move forward from there. The left-hand side can be expressed in the form of a dot-product as well. This isn't math homework, it's biological research - so my apologies if the framing isn't particularly clear.

Thanks in advance for your ideas and comments.