Quaternion Addition and Multiplication

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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, , , or ; you can also negate the red or blue quaternions.


Contributed by: Jon Perry (November 2011)
Open content licensed under CC BY-NC-SA



Write a quaternion as a scalar plus a three-vector, .

Quaternion addition is component-wise: .

Quaternion multiplication is defined by , where . is the vector dot product and is the vector cross product.

Snapshot 1: , but

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