A unit quaternion has the cosine of half the rotation angle as its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis ( here assumed fixed ) as its pseudovector ( or axial vector ) part.
22.
Here they are applied to relate the axial vector created as the cross product to the bivector-valued exterior product of two polar ( that is, not axial ) vectors and; the two products can be written as determinants expressed in the same way:
23.
For example, the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector.
24.
Before the advent of the electroweak theory and the Standard Model, George Sudarshan and Robert Marshak, and also independently Richard Feynman and Murray Gell-Mann, were able to determine the correct tensor structure ( vector minus axial vector, ) of the four-fermion interaction.
