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Orthogonal Transformations

Source lecture(s): PHY621 Lec1-2

Intuition

Rotations and reflections preserve lengths and angles—they are Euclidean symmetries.

Formal Definition

An orthogonal matrix \(R\) satisfies \(R^TR=RR^T=I\), so \(R^{-1}=R^T\).

Mathematical Formulation

\[R_z(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}\]

Derivation

The determinant condition preserves volume. In 3D, Rodrigues' formula builds any rotation from axis \(\hat{n}\) and angle \(\theta\).

Worked Example

\(90^\circ\) rotation about \(z\) sends \((1,0,0)\mapsto(0,1,0)\).

Common Mistakes

  • Assuming all orthogonal matrices are pure rotations (det \(=-1\) reflects).
  • Using Euler angles without checking ranges.

Quiz

Q1. What is \(\det R\) for a proper rotation?

Answer

\(+1\).