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

## 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 $$v$$.