# Thornton & Marion, Classical Dynamics, Fifth 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

1) Calculation of a determinant 2) Properties of the orthogonal transformation matrix

##### Solution

Since the transformation matrix isthen determinant of this matrix is So From Equation (1.13), which is the orthogonality condition, Therefore,