## 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.

### Proof

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

## 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 \(v\).