2D & 3D Rotations: From Complex Numbers to Quaternions

Why does multiplying by i rotate a point 90°? And why does that trick - which works so beautifully in 2D - completely fall apart the moment you try to extend it into 3D? This video follows that question all the way from complex numbers to quaternions: the strange four-dimensional numbers Hamilton spent a decade fighting for. It ends by showing exactly where they're running right now - inside the flight computer of every aircraft in the sky. 00:00 2D rotations via complex numbers — the core idea, polar form, and why z·e^{iθ} rotates 03:39 Why the obvious 3D extension breaks 04:20 The quaternion algebra — i² = j² = k² = ijk = −1 08:55 Deriving the 3D rotation formula — the "sandwich" v′ = qvq⁻¹ 19:55 Why quaternions beat Euler angles and rotation matrices — gimbal lock 20:31 Real-world payoff — quaternions flying an aircraft 25:06 Summary: 2D vs 3D, side by side If you've ever wondered what complex numbers are actually "for," or why game engines, flight computers, and spacecraft all reach for quaternions instead of the matrices you learned in school - this one's for you. #Quaternions #ComplexNumbers #Mathematics #LinearAlgebra #2DRotations #3DRotations