# quatmultiply

Calculate product of two quaternions

## Syntax

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

## Description

example

````quatprod = quatmultiply(q,r)` calculates the quaternion product, `quatprod`, for two quaternions, `q` and `r`. Aerospace Toolbox uses quaternions that are defined using the scalar-first convention. NoteQuaternion multiplication is not commutative. ```

## Examples

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This example shows how to 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 = 1×4 0.5000 1.2500 1.5000 0.2500 ```

This example shows how to determine the product of a 1-by-4 quaternion with itself.

```q = [1 0 1 0]; mult = quatmultiply(q)```
```mult = 1×4 0 0 2 0 ```

This example shows how to determine the product of 1-by-4 with two 1-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 = 2×4 0.5000 1.2500 1.5000 0.2500 1.9000 1.1000 2.1000 -0.9000 ```

## Input Arguments

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First quaternion or set of quaternions, specified as an m-by-4 matrix or 1-by-4 quaternion. Each element must be real.

`q` must have its scalar number as the first column.

Data Types: `double` | `single`

Second quaternionor set of quaternions, specified as an m-by-4 matrix or 1-by-4 quaternion. Each element must be real.

`r` must have its scalar number as the first column.

Data Types: `double` | `single`

## Output Arguments

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Output quaternion product, returned as a m-by-4 matrix.

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### q and r

Input quaternions `q` and `r` have the form:

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

and

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

### quatprod

Output quaternion product `quatprod` has the form of

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

where

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

## References

[1] Stevens, Brian L., Frank L. Lewis. Aircraft Control and Simulation, 2nd Edition. Hoboken, NJ: John Wiley & Sons, 2003.