Quaternion Addition and Multiplication

red quaternion components

scalar

1.5

i

1

j

1

k

1

i

j

k

-

blue quaternion components

scalar

-0.5

i

1

j

-1

k

0

1

i

j

k

-

sum

product

axes

1.5 + {1,1,1} + -0.5 + {1,-1,0}

= 1. + {2,0,1}

The quaternions are a number system with a noncommutative multiplication denoted here by *. They can be represented in various ways: as pairs of complex numbers, as four-dimensional vectors with real components, or as the sum of a real scalar and a real three-dimensional vector, as is done in this Demonstration. The scalar part of the quaternion is shown on a line and the vector part is shown in 3D.

Vary the red and blue quaternions to see the effect on their sum (orange) or product (green). Click a button to set a quaternion to either 1,

i

,

j

, or

k

; you can also negate the red or blue quaternions.