[Undergraduate Seminar] 2025-2 Semester – 1st Seminar “The Relationship Between Cross Product, Rotation, and Quaternions”
In the second semester of 2025, the Department of Mathematics hosted a student-initiated seminar titled “The Relationship Between Cross Product, Rotation, and Quaternions.” The event was organized by undergraduate students eager to explore mathematical concepts in greater depth, bringing together participants from diverse academic backgrounds at the Math Lounge on the 4th floor of the Mathematical Sciences Building. The seminar took place on Friday, November 14, at 5 p.m., with mathematics majors Hyeonseong Kim, Minjoon Kim, Kwangwoon Baek, Dohyun Lee, and Jungin Kim in attendance. They were joined by Jaewon Yu (Department of Computer Science), who sought to connect mathematical structures with computer graphics and geometric computation, and Taewoo Jung (Department of Physics), who was interested in the link between physical rotation and angular momentum. Focusing on the cross product defined in three-dimensional vector space, the seminar aimed to show that it is far more than a mere computational tool—it is deeply connected to the physical and geometric phenomenon of rotation. For example, the cross product of displacement and velocity vectors leads to angular velocity, and that of position and momentum vectors describes angular momentum. Likewise, the curl operator, which expresses the rotational component of a vector field, arises from combining the gradient operator with the cross product. Although students often encounter these ideas in coursework, grasping their structural meaning can be challenging. To provide a more fundamental understanding, the presenters introduced the perspective of interpreting the cross product as “the derivative of the square root of rotation.” This view highlights that the cross product is not a purely mechanical operation defined by the right-hand rule, but rather a natural construction emerging from the structure of rotational transformations and symmetry groups. Through this lens, participants deeply discussed why the cross product has its particular form only in three dimensions and how the quaternion structure can be used to represent rotations. As time allowed, the presenters also briefly introduced the concepts of Lie groups and Lie algebras related to rotation groups. Although these are typically treated at the graduate level, they arose naturally in the discussion as participants explored the essence of the cross product and rotation. Students from various majors shared their perspectives, finding that mathematical structures connect across disciplines through real-world phenomena, computer graphics, and engineering computations. The seminar was especially meaningful in that it was entirely student-led. It provided valuable experience for participants to explore mathematical ideas proactively, share their insights with peers, and broaden their understanding. One student shared, “I came to see the cross product not just as a computational formula but as something that embodies the structural concept of rotation. It was very meaningful to connect it with my own field of study.” The Department of Mathematics plans to continue actively supporting student-led academic endeavors. This seminar served as an inspiring example of an academic community that learns and grows together across disciplinary boundaries.
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