mtimes, *

Quaternion multiplication

Syntax

quatC = A*B

Description

quatC = A*B implements quaternion multiplication if either A or B is a quaternion. Either A or B must be a scalar.

You can use quaternion multiplication to compose rotation operators:

To compose a sequence of frame rotations, multiply the quaternions in the order of the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the order pq. The rotation operator becomes ${(p q)}^{*} v (p q)$ , where v represents the object to rotate specified in quaternion form. * represents conjugation.
To compose a sequence of point rotations, multiply the quaternions in the reverse order of the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the reverse order, qp. The rotation operator becomes $(q p) v {(q p)}^{*}$ .

example

Examples

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Multiply Quaternion Scalar and Quaternion Vector

Open Live Script

Create a 4-by-1 column vector, A, and a scalar, b. Multiply A times b.

A = quaternion(randn(4,4))

A = 4x1 quaternion array
      0.53767 +  0.31877i +   3.5784j +   0.7254k
       1.8339 -   1.3077i +   2.7694j - 0.063055k
      -2.2588 -  0.43359i -   1.3499j +  0.71474k
      0.86217 +  0.34262i +   3.0349j -  0.20497k

b = quaternion(randn(1,4))

b = quaternion
    -0.12414 +  1.4897i +   1.409j +  1.4172k

C = A*b

C = 4x1 quaternion array
      -6.6117 +   4.8105i +  0.94224j -   4.2097k
      -2.0925 +   6.9079i +   3.9995j -   3.3614k
       1.8155 -   6.2313i -    1.336j -     1.89k
      -4.6033 +   5.8317i + 0.047161j -    2.791k

Input Arguments

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`A` — Input to multiply
`quaternion` object | array of `quaternion` objects | real scalar | array of real numbers

Input to multiply, specified as a quaternion object, an array of quaternion objects of any dimensionality, a real scalar, or an array of real numbers of any dimensionality. Numeric values must be of data type single or double.

If B is nonscalar, then A must be scalar.

`B` — Input to multiply
`quaternion` object | array of `quaternion` objects | real scalar | array of real numbers

If A is nonscalar, then B must be scalar.

Output Arguments

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`quatC` — Quaternion product
`quaternion` object | array of `quaternion` objects

Quaternion product, returned as a quaternion object or an array of quaternion objects.

Algorithms

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Quaternion Multiplication by a Real Scalar

Given a quaternion

$q = a_{q} + b_{q} i + c_{q} j + d_{q} k,$

the product of q and a real scalar β is

$β q = β a_{q} + β b_{q} i + β c_{q} j + β d_{q} k$

Quaternion Multiplication by a Quaternion Scalar

The definition of the basis elements for quaternions,

$i^{2} = j^{2} = k^{2} = ijk = - 1,$

can be expanded to populate a table summarizing quaternion basis element multiplication:

	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

When reading the table, the rows are read first, for example: ij = k and ji = −k.

Given two quaternions, $q = a_{q} + b_{q} i + c_{q} j + d_{q} k,$ and $p = a_{p} + b_{p} i + c_{p} j + d_{p} k$ , the multiplication can be expanded as:

$\begin{array}{l} z = p q = (a_{p} + b_{p} i + c_{p} j + d_{p} k) (a_{q} + b_{q} i + c_{q} j + d_{q} k) \\ = a_{p} a_{q} + a_{p} b_{q} i + a_{p} c_{q} j + a_{p} d_{q} k \\ + b_{p} a_{q} i + b_{p} b_{q} i^{2} + b_{p} c_{q} ij + b_{p} d_{q} ik \\ + c_{p} a_{q} j + c_{p} b_{q} ji + c_{p} c_{q} j^{2} + c_{p} d_{q} jk \\ + d_{p} a_{q} k + d_{p} b_{q} ki + d_{p} c_{q} kj + d_{p} d_{q} k^{2} \end{array}$

You can simplify the equation using the quaternion multiplication table:

$\begin{array}{l} z = p q = a_{p} a_{q} + a_{p} b_{q} i + a_{p} c_{q} j + a_{p} d_{q} k \\ + b_{p} a_{q} i - b_{p} b_{q} + b_{p} c_{q} k - b_{p} d_{q} j \\ + c_{p} a_{q} j - c_{p} b_{q} k - c_{p} c_{q} + c_{p} d_{q} i \\ + d_{p} a_{q} k + d_{p} b_{q} j - d_{p} c_{q} i - d_{p} d_{q} \end{array}$

References

[1] Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton, NJ: Princeton University Press, 2007.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2018a

mtimes, *

Syntax

Description

Examples

Multiply Quaternion Scalar and Quaternion Vector

Input Arguments

`A` — Input to multiply
`quaternion` object | array of `quaternion` objects | real scalar | array of real numbers

`B` — Input to multiply
`quaternion` object | array of `quaternion` objects | real scalar | array of real numbers

Output Arguments

`quatC` — Quaternion product
`quaternion` object | array of `quaternion` objects

Algorithms

Quaternion Multiplication by a Real Scalar

Quaternion Multiplication by a Quaternion Scalar

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

See Also

Functions

Objects

mtimes, *

Syntax

Description

Examples

Multiply Quaternion Scalar and Quaternion Vector

Input Arguments

A — Input to multiply quaternion object | array of quaternion objects | real scalar | array of real numbers

B — Input to multiply quaternion object | array of quaternion objects | real scalar | array of real numbers

Output Arguments

quatC — Quaternion product quaternion object | array of quaternion objects

Algorithms

Quaternion Multiplication by a Real Scalar

Quaternion Multiplication by a Quaternion Scalar

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

Version History

See Also

Functions

Objects

`A` — Input to multiply
`quaternion` object | array of `quaternion` objects | real scalar | array of real numbers

`B` — Input to multiply
`quaternion` object | array of `quaternion` objects | real scalar | array of real numbers

`quatC` — Quaternion product
`quaternion` object | array of `quaternion` objects

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.