# quatmultiply

Calculate product of two quaternions

## Syntax

```n = quatmultiply(q,r) ```

## Description

`n = quatmultiply(q,r)` calculates the quaternion product, `n`, for two given quaternions, `q` and `r`. Inputs `q` and `r` can each be either an `m`-by-4 matrix containing `m` quaternions, or a single 1-by-4 quaternion. `n` returns an `m`-by-4 matrix of quaternion products. Each element of `q` and `r` must be a real number. Additionally, `q` and `r` have their scalar number as the first column.

The quaternions have the form of

`$q={q}_{0}+i{q}_{1}+j{q}_{2}+k{q}_{3}$`

and

`$r={r}_{0}+i{r}_{1}+j{r}_{2}+k{r}_{3}$`

The quaternion product has the form of

`$n=q×r={n}_{0}+i{n}_{1}+j{n}_{2}+k{n}_{3}$`

where

`$\begin{array}{l}{n}_{0}=\left({r}_{0}{q}_{0}-{r}_{1}{q}_{1}-{r}_{2}{q}_{2}-{r}_{3}{q}_{3}\right)\\ {n}_{1}=\left({r}_{0}{q}_{1}+{r}_{1}{q}_{0}-{r}_{2}{q}_{3}+{r}_{3}{q}_{2}\right)\\ {n}_{2}=\left({r}_{0}{q}_{2}+{r}_{1}{q}_{3}+{r}_{2}{q}_{0}-{r}_{3}{q}_{1}\right)\\ {n}_{3}=\left({r}_{0}{q}_{3}-{r}_{1}{q}_{2}+{r}_{2}{q}_{1}+{r}_{3}{q}_{0}\right)\end{array}$`

### Note

Quaternion multiplication is not commutative.

## Examples

Determine the product of two 1-by-4 quaternions:

```q = [1 0 1 0]; r = [1 0.5 0.5 0.75]; mult = quatmultiply(q, r) mult = 0.5000 1.2500 1.5000 0.2500```

Determine the product of a 1-by-4 quaternion with itself:

```q = [1 0 1 0]; mult = quatmultiply(q) mult = 0 0 2 0```

Determine the product of 1-by-4 and 2-by-4 quaternions:

```q = [1 0 1 0]; r = [1 0.5 0.5 0.75; 2 1 0.1 0.1]; mult = quatmultiply(q, r) mult = 0.5000 1.2500 1.5000 0.2500 1.9000 1.1000 2.1000 -0.9000```

## References

[1] Stevens, Brian L., Frank L. Lewis, Aircraft Control and Simulation, Wiley–Interscience, 2nd Edition.