# power, .^

Element-wise quaternion power

## Syntax

``C = A.^b``

## Description

example

````C = A.^b` raises each element of `A` to the corresponding power in `b`. ```

## Examples

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Create a quaternion and raise it to a real scalar power.

`A = quaternion(1,2,3,4)`
```A = quaternion 1 + 2i + 3j + 4k ```
```b = 3; C = A.^b```
```C = quaternion -86 - 52i - 78j - 104k ```

Create a 2-by-1 quaternion array and raise it to powers from a 2-D array.

`A = quaternion([1:4;5:8])`
```A=2×1 quaternion array 1 + 2i + 3j + 4k 5 + 6i + 7j + 8k ```
`b = [1 0 2; 3 2 1]`
```b = 2×3 1 0 2 3 2 1 ```
`C = A.^b`
```C=2×3 quaternion array 1 + 2i + 3j + 4k 1 + 0i + 0j + 0k -28 + 4i + 6j + 8k -2110 - 444i - 518j - 592k -124 + 60i + 70j + 80k 5 + 6i + 7j + 8k ```

## Input Arguments

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Base, specified as a scalar, vector, matrix, or multidimensional array.

Data Types: `quaternion` | `single` | `double`

Exponent, specified as a real scalar, vector, matrix, or multidimensional array.

Data Types: `single` | `double`

## Output Arguments

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Each element of quaternion A raised to the corresponding power in b, returned as a scalar, vector, matrix, or multidimensional array.

Data Types: `quaternion`

## Algorithms

The polar representation of a quaternion $A=a+b\text{i}+c\text{j}+d\text{k}$ is given by

`$A=‖A‖\left(\mathrm{cos}\theta +\stackrel{^}{u}\mathrm{sin}\theta \right)$`

where θ is the angle of rotation, and û is the unit quaternion.

Quaternion A raised by a real exponent b is given by

`$P=A.^b={‖A‖}^{b}\left(\mathrm{cos}\left(b\theta \right)+\stackrel{^}{u}\mathrm{sin}\left(b\theta \right)\right)$`

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