# Harmonic vector realm

This post expands on Harmonic Algebra posted here.

A vector space, or better a vector realm, to avoid connecting it with physical space, is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.

The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum u + v belongs to V.
(1) Commutative law: For all vectors u and v in V, u + v = v + u
(2) Associative law: For all vectors u, v, w in V, u + (v + w) = (u + v) + w
(3) Additive identity: The set V contains an additive identity element, usually denoted by 0, such that for any vector v in V, 0 + vv and v + 0 = v.
(4) Additive inverses: For any vector v in V, the equations v + x = 0 and x + v = 0 have a solution x in V, called an additive inverse of v, and denoted by −v.

The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:

Closure: If v is any vector in V, and c is any real number, then the product c · v belongs to V.
(5) Distributive law: For all real numbers c and all vectors uv in V, c · (u + v) = c · u + c · v
(6) Distributive law: For all real numbers c, d and all vectors v in V, (c + d) · v = c · v + d · v
(7) Associative law: For all real numbers c, d and all vectors v in V, c · (d · v) = (cd) · v
(8) Multiplicative identity: The set V contains a multiplicative identity element, usually denoted by 1, such that for any vector v in V, 1 · v = v

Consider the non-zero real numbers together with the element ∞ as components of Euclidean vectors, with · as the usual dot product, and vector addition defined as harmonic addition:

$\mathbf{u}\boxplus&space;\mathbf{v}=\left&space;[&space;\mathbf{u}^{-1}+\mathbf{v}^{-1}&space;\right&space;]^{-1}$

which is undefined for zero vectors, but has the additive identity ∞ (infinity). It is isomorphic to the vector space (or realm) with 0 as the additive identity.

The independent variable is usually in the denominator but if the independent variable is in the numerator, then the denominator contains a dependent variable. See here for what this looks like. The addition of quotients with a dependent vector in the denominator follows the above.