Is there a variant of the dot-product operation that returns $\frac{a_1}{b_1} + \frac{a_2}{b_2}$ from vectors $[a_1,a_2]$ and $[b_1, b_2]$?

Question

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 $$\frac{a_1}{b_1} + \frac{a_2}{b_2}$$ 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.

Answer 1

Probably not: the dot product has a geometric meaning (the product of the lengths of the vectors times the cosine of the angle between them): it's a scalar, i.e. a number that does not change under orthogonal coordinate transformations ("it's the same number in every coordinate system").

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.

Note that e.g. the x-coordinate of a vector is such a number: it is not the same in every coordinate system. It can nevertheless be useful. So there might be a use of your number somewhere, but if so, I don't know about it.

Answer 2

To answer the asked question: NO. While there are many inverse simplification formulas that exist, none can be resolved to fulfill your requested outcome for all input values.

---

I agree that the question as asked has an infinite number of solutions, but the infinite solutions are bounded by the constraints of the equations and any input constraints.

For a carrier wave(sin or cos) input 'B' modulated by 4 base-band signals 'A' is the phase aligned dot product of A=[A1,A2,A3,A4] dot B.
Given the prior knowledge of the waveform B and 4 modulated base-band signals exist, it is possible to reconstruct(i.e. inverse dot product) the original waveforms from A=[A1,A2,A3,A4]. The imperfection is manifested in the phase shift, as the sign of A is unknowable.

---

The reason for this answer is a Google search for 'inverse dot product' shows this question as the poster child of impossibility, without exploring any of the input conditions that could make it possible.