# quatdivide

Divide quaternion by another quaternion

## Syntax

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

## Description

`n = quatdivide(q,r)` calculates the result of quaternion division, `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 quotients. 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 resulting quaternion from the division has the form of

`$t=\frac{q}{r}={t}_{0}+i{t}_{1}+j{t}_{2}+k{t}_{3}$`

where

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

## Examples

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

```q = [1 0 1 0]; r = [1 0.5 0.5 0.75]; d = quatdivide(q, r) d = 0.7273 0.1212 0.2424 -0.6061```

Determine the division of a 2-by-4 quaternion by a 1-by-4 quaternion:

```q = [1 0 1 0; 2 1 0.1 0.1]; r = [1 0.5 0.5 0.75]; d = quatdivide(q, r) d = 0.7273 0.1212 0.2424 -0.6061 1.2727 0.0121 -0.7758 -0.4606```

## References

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