This post is based on research papers by *Anderson and Trapp*, *Berlinet*, and the post on *Reciprocal arithmetic*.

The vector inverse **x**^{−1} is defined as

with positive norm. For a non-zero scalar *k*,

The reciprocal (or harmonic or parallel) sum is symbolized in various ways, but I prefer a “boxplus” to maintain its relation with addition. The reciprocal sum of vectors **x** and **y** is defined as

if **x** ≠ **0**, **y** ≠ **0**, and **x** + **y** ≠ **0**; otherwise the sum equals **0**.

The reciprocal sum is commutative and associative, among other properties. Given two vectors with **x** + **y** ≠ **0** and non-zero scalar *k*,

The arithmetic mean for vectors **x** and **y** is

The reciprocal difference of vectors **x** and **y** such that **x** + **y** ≠ **0** is defined as

The harmonic mean for vectors **x** and **y** is