Reflection in a line is represented by a sandwiching construction involving the geometric product. Though that may have seemed a curiosity in the previous chapter, we will show that it is crucial to the representation of operators in geometric algebra. Geometrically, all orthogonal transformations can be considered as multiple reflections. Algebraically, this leads to their representation as a geometric product of unit vectors.

An even number of reflections gives a rotation, represented as a rotor—the geometric product of an even number of unit vectors. We show that rotors encompass and extend complex numbers and quaternions, and present a real 3-D visualization of the quaternion product. Rotors transcend ...

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