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