Dual Quaternions and Dual Quaternion Vectors

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 show that the sum of the product of a quaternion and the conjugate of another quaternion, and the product of the other quaternion and the conjugate of that quaternion, is a real number. 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.