# 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.