A Geometric Interpretation of the Product of Two Quaternions

  If we multiply two vector quaternions we get a third quaterion which we can decompose into scalar and vector parts. We can rewrite the vector part as the product of a unit vector quaternion and a scalar magnitude. The two magnitudes turn out to be equal to the product the magnitudes of the original quaternions and either a cosine or sine factor.

This suggests the interpretation of the magnitudes of the two parts of the quaternions as the area of two "complementary" parallelograms. The upper parallelogram below is corresponds to the magnitude of the vector part of the product ba or absinθ. The lower parallelogram has the complement of angle θ between a and b' and the sine of this equals cosθ so its product is abcosθ, the magnitudes b and b' being equal.