Delve into the practical implementation of quaternions for describing the algebra of rotations in three-dimensional space in this hour-long mathematics lecture. Explore the concept of half-turns as a key to connecting with the algebraic structure of quaternions, and discover the foundations of Vector Trigonometry. Learn how to decompose quaternion multiplication into its action on perpendicular planes, gaining insight into the geometry of this operation. Understand the relationship between quaternion multiplication and rotations in space, providing a deeper comprehension of this fundamental mathematical concept. This lecture offers a thorough examination of quaternions and their applications in spatial rotations, building on previous discussions in the series.