Quaternion Addition and Multiplication

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.


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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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