Dual Quaternions and Dual Quaternion Vectors

TL;DR

This paper extends the mathematical framework of dual numbers and quaternions by defining an order, absolute value, and vector norms for dual quaternion vectors, enhancing their algebraic and geometric utility.

Contribution

It introduces a total order, an absolute value function, and extends common vector norms to dual quaternion vectors, providing new tools for their analysis.

Findings

Defined a total order for dual numbers

Introduced an absolute value function for dual numbers

Extended 1-norm, infinity-norm, and 2-norm to dual quaternion vectors

Abstract

We introduce a total order and the absolute value function for dual numbers. The absolute value function of dual numbers are with dual number values, and have properties similar to the properties of the absolute value function of real numbers. We define the magnitude of a dual quaternion, as a dual number. Based upon these, we extended $1$-norm, $\infty$-norm and $2$-norm to dual quaternion vectors.