Editing Quaternion (section)

=== Scalar and vector parts ===
A quaternion of the form {{math|''a'' + 0 '''i''' + 0 '''j''' + 0 '''k'''}}, where {{mvar|a}} is a real number, is called '''scalar''', and a quaternion of the form {{math|0 + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}}, where {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are real numbers, and at least one of {{mvar|b}}, {{mvar|c}}, or {{mvar|d}} is nonzero, is called a '''vector quaternion'''.  If {{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} is any quaternion, then {{mvar|a}} is called its '''scalar part''' and {{math|''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} is called its '''vector part'''. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the '''vector''' part as vectors in three-dimensional space.  With this convention, a vector is the same as an element of the vector space <math>\mathbb R^3.</math>{{efn|The vector part of a quaternion is an "axial" vector or "[[pseudovector]]", ''not'' an ordinary or "polar" vector, as was formally proved by Altmann (1986).<ref>{{cite book |first=S.L. |last=Altmann |at=Ch.&nbsp;12 |title=Rotations, Quaternions, and Double Groups}}</ref> A polar vector can be represented in calculations (for example, for rotation by a quaternion "similarity transform") by a pure imaginary quaternion, with no loss of information, but the two should not be confused. The axis of a "binary" (180°) rotation quaternion corresponds to the direction of the represented polar vector in such a case.}}

Hamilton also called vector quaternions '''right quaternions'''<ref>{{cite book |url=https://archive.org/details/bub_gb_fIRAAAAAIAAJ |title=Elements of Quaternions |publisher=Longmans, Green, & Company |article=Article 285 |page=[https://archive.org/details/bub_gb_fIRAAAAAIAAJ/page/n381 310] |author=Hamilton, Sir William Rowan |year=1866}}</ref><ref>{{cite journal |url=http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;q1=right%20quaternion;rgn=full%20text;idno=05140001;didno=05140001;view=image;seq=81 |author=Hardy |title=Elements of Quaternions |journal=Science |year=1881 |volume=2 |issue=75 |page=65 |publisher=library.cornell.edu|doi=10.1126/science.os-2.75.564 |pmid=17819877 }}</ref> and real numbers (considered as quaternions with zero vector part) '''scalar quaternions'''.

If a quaternion is divided up into a scalar part and a vector part, that is,

<math display=block>
\mathbf q = (r,\,\vec{v}),\ \mathbf q \in \mathbb{H},\ r \in \mathbb{R},\ \vec{v}\in \mathbb{R}^3,
</math>

then the formulas for addition, multiplication, and multiplicative inverse are

<math display=block>\begin{align}
(r_1,\,\vec{v}_1) + (r_2,\,\vec{v}_2)
&= (r_1 + r_2,\,\vec{v}_1 + \vec{v}_2), \\[5mu]
(r_1,\,\vec{v}_1) (r_2,\,\vec{v}_2)
&= (r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\,r_1\vec{v}_2+r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2), \\[5mu]
(r,\,\vec{v})^{-1}
&= \left(\frac{r}{r^2 + \vec{v}\cdot\vec{v}},\ \frac{-\vec{v}}{r^2 + \vec{v}\cdot\vec{v}}\right),
\end{align}</math>

where "<math>{}\cdot{}</math>" and "<math>\times</math>" denote respectively the [[dot product]] and the [[cross product]].