An Example Involving a Series of Rotations

  We can do a series of rotations to show that they do not affect the area of a parallelogram. We start with two vector quaternions that along with the origin form an equilateral triangle. The quaternion product gives us the area of the two parallelograms. Next we find the set of  projections vectors for the vectors to be rotated starting with a rotation of angle α about k̂. Next we compute the pole for a rotation of angle β in the â',k̂-plane. Finally we do a rotation of angle γ about â".

We first check the calculations done with the algebraic expressions with numerical calculations and compute the magnitude of the vector part of the product.

Doing the rotation of α about k̂ gives,

And a rotation of β about â'k̂ gives,

Finally the rotation of γ about â" gives,

Note that the magnitudes of the scalar and vector parts of the products are unchanged by the rotations. This method is rather cumbersome since we need the polar, normal and binormal parts for each vector rotated. Hamilton's half angle triple products do not need them and the same q̂ works for all points subject to the same rotation.