\( % Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}} \)

Quaternion sum

The sum of two quaternions \(\group{w_1,v_1}\) and \(\group{w_2,v_2}\) is defined as a quaternion \(\group{w_1+w_2,v_1+v_2}\). The sum is associative i.e. \((x+y)+z = x+(y+z)\) holds and commutative i.e. \(x+y = y+x\) holds for any quaternions \(x\),\(y\), and \(z\).

Scalar quaternion product

The product of a scalar \(s\) and a quaternion \(\group{w,v}\) is defined as a quaternion \(\group{sw,sv}\).

Quaternion product

The product of two quaternions \(\group{w_1,v_1}\) and \(\group{w_2,v_2}\) is defined as a quaternion \(\group{w_1 w_2 - \dot{v_1}{v_2}, w_1 v_2 + w_2 v_1 + \cross{v_1}{v_2}}\).

Unit quaternion

A quaternion \(\group{1,\vec{0}}\) is called the unit quaternion.

The unit quaternion has the following properties:

  • \(\group{w,v} \group{1,\vec{0}} = \group{1,\vec{0}} \group{w,v}\) (Commutativity)
  • \(\group{w,v} \group{1,\vec{0}} = \group{w,v}\) (Multiplicative identity)

Proof

We proof \(\group{w,v} \group{1,\vec{0}} = \group{1,\vec{0}} \group{w,v}\) holds.

Quaternion product where \(\group{w_1,v_1}=\group{w,v}\) and \(\group{w_2,v_2}=\group{1,\vec{0}}\).
\[\begin{aligned} &=\group{w 1 - \dot{v}{\vec{0}}, \dot{w}{\vec{0}} + 1 v + \cross{v}{\vec{0}}} \end{aligned}\]
\[\begin{aligned} &=\group{w, v} \end{aligned}\]

Proof

We proof \(\group{w,v} \group{1,\vec{0}} = \group{w,v}\)

Quaternion product where =\(\group{w_1,v_1}=\group{1,\vec{0}}\) and \(\group{w_2,v_2}=\group{w,v}\).
\[\begin{aligned} &=\group{1 w - \dot{\vec{0}}{v}, 1 v + w \vec{0} + \cross{\vec{0}}{v}} \end{aligned}\]
\[\begin{aligned} &=\group{w, v} \end{aligned}\]

Quaternion dot product

The dot product of two quaternions \(\group{w_1,v_1}\) and \(\group{w_2,v_2}\) is defined as a scalar \(w_1 w_2 + \dot{v_1}{v_2}\).

Quaternion corresponding to a vector

\(q=\group{0,v}\) is the quaternion corresponding to a vector \(v\).