Editing Quaternion (section)

== Quaternions and three-dimensional geometry ==
The vector part of a quaternion can be interpreted as a coordinate vector in <math>\mathbb R^3;</math> therefore, the algebraic operations of the quaternions reflect the geometry of <math>\mathbb R^3.</math> Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics.<ref name="Shoemake"/>

For the remainder of this section, {{math|'''i'''}}, {{math|'''j'''}}, and {{math|'''k'''}} will denote both the three imaginary<ref>{{cite book |url=https://archive.org/details/vectoranalysisa00wilsgoog |quote=right tensor dyadic |title=Vector Analysis |publisher=Yale University Press |last1=Gibbs |first1=J. Willard |last2=Wilson |first2= Edwin Bidwell |year=1901 |page=[https://archive.org/details/vectoranalysisa00wilsgoog/page/n452 428]}}</ref> basis vectors of <math>\mathbb H</math> and a basis for <math>\mathbb R^3.</math> Replacing {{math|'''i'''}} by {{math|−'''i'''}}, {{math|'''j'''}} by {{math|−'''j'''}}, and {{math|'''k'''}} by {{math|−'''k'''}} sends a vector to its [[additive inverse]], so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the ''spatial inverse''.

For two vector quaternions {{nowrap|{{math|''p'' {{=}} ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}} }} and {{nowrap|{{math|''q'' {{=}} ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}} }} their [[dot product]], by analogy to vectors in <math>\mathbb R^3,</math> is

<math display=block>p \cdot q = b_1 b_2 + c_1 c_2 + d_1 d_2.</math>

It can also be expressed in a component-free manner as

<math display=block>p \cdot q = \textstyle\frac{1}{2}(p^*q + q^*p) = \textstyle\frac{1}{2}(pq^* + qp^*).</math>

This is equal to the scalar parts of the products {{math|''pq''<sup>∗</sup>, ''qp''<sup>∗</sup>, ''p''<sup>∗</sup>''q'', and ''q''<sup>∗</sup>''p''}}. Note that their vector parts are different.

The [[cross product]] of {{mvar|p}} and {{mvar|q}} relative to the orientation determined by the ordered basis {{math|'''i''', '''j'''}}, and {{math|'''k'''}} is

<math display="block">p \times q = (c_1 d_2 - d_1 c_2)\mathbf i + (d_1 b_2 - b_1 d_2)\mathbf j + (b_1 c_2 - c_1 b_2)\mathbf k.</math>

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product {{math|''pq''}} (as quaternions), as well as the vector part of {{math|−''q''<sup>∗</sup>''p''<sup>∗</sup>}}. It also has the formula

<math display=block>p \times q = \textstyle\tfrac{1}{2}(pq - qp).</math>

For the [[Commutator#Ring theory|commutator]], {{math|[''p'', ''q''] {{=}} ''pq'' − ''qp''}}, of two vector quaternions one obtains

<math display=block>[p,q]= 2p \times q,</math>

{{anchor|commutation relationship}}which gives the commutation relationship

<math display=block>qp= pq - 2p \times q.</math>

In general, let {{mvar|p}} and {{mvar|q}} be quaternions and write

<math display=block>\begin{align}
p &= p_\text{s} + p_\text{v}, \\[5mu]
q &= q_\text{s} + q_\text{v},
\end{align}</math>

where {{math|''p''<sub>s</sub>}} and {{math|''q''<sub>s</sub>}} are the scalar parts, and {{math|''p''<sub>v</sub>}} and {{math|''q''<sub>v</sub>}} are the vector parts of {{mvar|p}} and {{mvar|q}}. Then we have the formula

<math display="block">pq = (pq)_\text{s} + (pq)_\text{v} = (p_\text{s}q_\text{s} - p_\text{v}\cdot q_\text{v}) + (p_\text{s} q_\text{v} + q_\text{s} p_\text{v}  + p_\text{v} \times q_\text{v}).</math>

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of vector quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear. Hamilton<ref name="Hamilton">{{cite journal |author-link=William Rowan Hamilton |first=W.R. |last=Hamilton |year=1844–1850 |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/ |title=On quaternions or a new system of imaginaries in algebra |journal=[[Philosophical Magazine]] |department=David R. Wilkins collection |publisher=[[Trinity College Dublin]]}}</ref> showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in [[Elliptic geometry]].

Unit quaternions can be identified with rotations in <math>\mathbb R^3</math> and were called [[versor]]s by Hamilton.<ref name="Hamilton" /> Also see [[Quaternions and spatial rotation]] for more information about modeling three-dimensional rotations using quaternions.

See [[Andrew J. Hanson|Hanson]] (2005)<ref>{{cite web |url=http://www.cs.indiana.edu/~hanson/quatvis/ |title=Visualizing Quaternions |publisher=Morgan-Kaufmann/Elsevier |year=2005}}</ref> for visualization of quaternions.