# Is there an “inverse” of the dot-product?

The dot product of $$[a_1,a_2]$$ and $$[b_1, b_2]$$ is $$a_1b_1 + a_2b_2$$ (and so on for bigger vectors). What I'm wondering is if there's any definition of a function s.t. the "invdot" product of $$[a_1,a_2]$$ and $$[b_1, b_2]$$ is $$\frac{a_1}{b_1} + \frac{a_2}{b_2}$$, and if so, are there any other properties/derivations of this function from the dot product? I can see it's the same as taking the reciprocal of all the components of $$b$$ and then doing the dot product on that, but I can't understand how I'd do that in my specific use case.

For context, I'm trying to solve a problem where I want to algebraically manipulate $$a_1b_1 + a_2b_2$$ to $$\frac{a_1}{b_1} + \frac{a_2}{b_2}$$, and think that this might help if it's a well known operation.

• Note that the dot product sends two vectors to a scalar, the inverse of that won't make much sense (there are infinitely many such mappings). – lightxbulb Jan 23 '19 at 22:54
• The dot product is not injective, so there cannot be an inverse. However, maybe you used the term "inverse" by mistake, since you are talking about reciprocals. – Ben W Jan 23 '19 at 22:59
• It's "inverse" as in "opposite", not "reverse". It's not the common meaning in mathematics, but it's pedantic to require everyone here to adhere to such standards. Particularly when they're experimenting. – Arthur Jan 23 '19 at 23:03
• Yep, +Arthur's got it - though I see where you guys are coming too. – Robin Aldabanx Jan 23 '19 at 23:07

Your operation (it is not an "inverse" of the dot product, so let's not call it that) takes two vectors and produces a number, but the number is not invariant under coordinate transformations (e.g. it becomes infinite when you take a coordinate system where $$\vec{b}$$ is along the x-axis): there is no geometric meaning that can be assigned to it, because it is not the same in every coordinate system.