Classical Mechanics — 2.5: Vector Components Under Coordinate Rotation

Full course — free exercises, Feynman reviews, and AI-graded feedback: https://ludium.ai/courses/classical-m... When you rotate a coordinate system by angle θ, a vector's components change — but the vector itself does not. This video derives the exact transformation equations relating components in two rotated frames, starting from first principles with unit vector decomposition. A worked numerical example and a magnitude invariance proof show why these equations matter throughout physics. Key concepts covered: • Passive rotation: the vector is fixed, only the coordinate axes rotate • Decomposing rotated unit vectors î' and ĵ' in the original basis • Derivation of the transformation law: x' = x cos θ + y sin θ, y' = −x sin θ + y cos θ • Why the minus sign appears in y' but not in x' • Independent verification using angle-addition identities • Worked example: computing primed components for a vector at θ = 60° • Magnitude invariance: |r| is identical in every rotated frame • This transformation law as the prototype for rotation matrices and tensors ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from https://ocw.mit.edu/courses/8-01sc-cl...