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# quatrotate

Rotate vector by quaternion

## Syntax

n = quatrotate(q,r)

## Description

n = quatrotate(q,r) calculates the rotated vector, n, for a quaternion, q, and a vector, r. q is either an m-by-4 matrix containing m quaternions, or a single 1-by-4 quaternion. r is either an m-by-3 matrix, or a single 1-by-3 vector. n returns an m-by-3 matrix of rotated vectors. Each element of q and r must be a real number. Additionally, q has its scalar number as the first column.

The quaternion has the form of

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

The vector has the form of

$v=i{v}_{1}+j{v}_{2}+k{v}_{3}$

The rotated vector has the form of

${v}^{\prime }=\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right]=\left[\begin{array}{ccc}\left(1-2{q}_{2}^{2}-2{q}_{3}^{2}\right)& 2\left({q}_{1}{q}_{2}+{q}_{0}{q}_{3}\right)& 2\left({q}_{1}{q}_{3}-{q}_{0}{q}_{2}\right)\\ 2\left({q}_{1}{q}_{2}-{q}_{0}{q}_{3}\right)& \left(1-2{q}_{1}^{2}-2{q}_{3}^{2}\right)& 2\left({q}_{2}{q}_{3}+{q}_{0}{q}_{1}\right)\\ 2\left({q}_{1}{q}_{3}+{q}_{0}{q}_{2}\right)& 2\left({q}_{2}{q}_{3}-{q}_{0}{q}_{1}\right)& \left(1-2{q}_{1}^{2}-2{q}_{2}^{2}\right)\end{array}\right]\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right]$

## Examples

Rotate a 1-by-3 vector by a 1-by-4 quaternion:

```q = [1 0 1 0];
r = [1 1 1];
n = quatrotate(q, r)

n =

-1.0000    1.0000    1.0000```

Rotate a 1-by-3 vector by a 2-by-4 quaternion:

```q = [1 0 1 0; 1 0.5 0.3 0.1];
r = [1 1 1];
n = quatrotate(q, r)

n =

-1.0000    1.0000    1.0000
0.8519    1.4741    0.3185```

Rotate a 2-by-3 vector by a 1-by-4 quaternion:

```q = [1 0 1 0];
r = [1 1 1; 2 3 4];
n = quatrotate(q, r)

n =

-1.0000    1.0000    1.0000
-4.0000    3.0000    2.0000```

Rotate a 2-by-3 vector by a 2-by-4 quaternion:

```q = [1 0 1 0; 1 0.5 0.3 0.1];
r = [1 1 1; 2 3 4];
n = quatrotate(q, r)

n =

-1.0000    1.0000    1.0000
1.3333    5.1333    0.9333```

## References

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