🎬Algebraic Cross Product vs. Geometric Lie Bracket
The contrast between the two animations reveals a fundamental distinction in mathematical physics: while the Cross Product is a static, algebraic operation identifying a radial direction orthogonal to the two input vectors, the Lie Bracket is a dynamic, differential operation measuring the "drift" caused by non-commuting flows. The animation of the Commutator Loop demonstrates that following these rotations in sequence fails to return to the starting point, physically manifesting the Lie Bracket as the gap left behind. Ultimately, this illustrates that while the cross product describes the pointwise geometry of the arrows (pointing outward), the Lie Bracket describes the algebraic structure of the rotation group SO(3), proving that the interaction of two rotations generates a third rotation rather than radial movement.