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## 9.64 Quaternion

The domain constructor Quaternion implements quaternions over commutative rings. For information on related topics see ComplexXmpPage and OctonionXmpPage . You can also issue the system command )show Quaternion to display the full list of operations defined by Quaternion.

The basic operation for creating quaternions is quatern quatern Quaternion . This is a quaternion over the rational numbers.

q := quatern(2/11,-8,3/4,1)

2 11 - 8 i + 3 4 j + k

Type: Quaternion Fraction Integer

The four arguments are the real part, the i imaginary part, the j imaginary part, and the k imaginary part, respectively.

[real q, imagI q, imagJ q, imagK q]

[ 2 11 , - 8 , 3 4 , 1 ]

Type: List Fraction Integer

Because q is over the rationals (and nonzero), you can invert it.

inv q

352 126993 + 15488 126993 i - 484 42331 j - 1936 126993 k

Type: Quaternion Fraction Integer

The usual arithmetic (ring) operations are available

q**6

- 2029490709319345 7256313856 - 48251690851 1288408 i + 144755072553 41229056 j + 48251690851 10307264 k

Type: Quaternion Fraction Integer
r := quatern(-2,3,23/9,-89); q + r

- 20 11 - 5 i + 119 36 j - 88 k

Type: Quaternion Fraction Integer

In general, multiplication is not commutative.

q * r - r * q

- 2495 18 i - 1418 j - 817 18 k

Type: Quaternion Fraction Integer

There are no predefined constants for the imaginary i, j, and k parts, but you can easily define them.

i:=quatern(0,1,0,0); j:=quatern(0,0,1,0); k:=quatern(0,0,0,1)

k

Type: Quaternion Integer

These satisfy the normal identities.

[i*i, j*j, k*k, i*j, j*k, k*i, q*i]

[ - 1 , - 1 , - 1 , k , i , j , 8 + 2 11 i + j - 3 4 k ]

Type: List Quaternion Fraction Integer

The norm is the quaternion times its conjugate.

norm q

126993 1936

Type: Fraction Integer
conjugate q

2 11 + 8 i - 3 4 j - k

Type: Quaternion Fraction Integer
q * %

126993 1936

Type: Quaternion Fraction Integer
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