# Thornton & Marion, Classical Dynamics, 5th Edition

## Chapter 1. Matrices, Vectors, and Vector Calculus

### Problem 05. The determinant of the transformation matrix

#### The problem asks you to

- show that

#### This problem assumes

- The transformation matrix to be a two-dimensional orthogonal matrix.

#### You should know about

- Calculation of a determinant
- Properties of the orthogonal transformation matrix

#### Solution

Since the transformation matrix is

So

From Equation (1.13), which is the orthogonality condition,

Therefore,