为方便理解四元数，首先针对两个相互平行或垂直的向量，定义它们之间的一种不可交换乘积，命名为格拉斯曼乘积，同时约定这一不可交换积满足分配律。由此，进一步给出任意两个向量之间格拉斯曼积的具体表达式，并引出四元数的概念和运算法则。从理论上证明，任意四元数都可表示为两个向量之间的格拉斯曼积，并可以利用单位四元数的正交变换来表示向量旋转的欧拉公式。"/>
Study on the Basic Theory of Quaternion and Eulerian Equation of the Rotating Vector" />
To understand the quaternion, this paper defines a non-exchange product between two vectors parallel or vertical to each other, names it as the Glassman product, and promises this non-exchange product meet the commutative law. Furthermore, this paper derives the specific expression of the Glassman product between any two vectors, and then leads to the conception of the quaternion and its operation rules. In theory, this paper proves that any quaternion can be represented as the Glassman product between two vectors, and the Eulerian equation of the rotating vector can be represented by the orthogonal transformation of the unit quaternion either."/>
<div style="line-height: 150%">Study on the Basic Theory of Quaternion and Eulerian Equation of the Rotating Vector</div>
Abstract To understand the quaternion, this paper defines a non-exchange product between two vectors parallel or vertical to each other, names it as the Glassman product, and promises this non-exchange product meet the commutative law. Furthermore, this paper derives the specific expression of the Glassman product between any two vectors, and then leads to the conception of the quaternion and its operation rules. In theory, this paper proves that any quaternion can be represented as the Glassman product between two vectors, and the Eulerian equation of the rotating vector can be represented by the orthogonal transformation of the unit quaternion either.
