# Thornton & Marion, Classical Dynamics, 5th Edition

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

### Problem 06. Another proof of the orthogonality condition

#### The problem asks you to

- prove that Equation (1.15) can be obtained by using the fact that the transformation matrix preserves the length of the line segment.

#### This problem assumes

- that the coordinate systems are both orthogonal.

#### We should know about

- Transformation (rotation) matrix
- Orthogonality condition

#### Solution

We assume a point is represented in the coordinate system by , and it can be also represented in the coordinate system by .

These coordinate systems have the same origin. The length of the line segment from the origin and the point is

Since the transformation matrix preserves the length of the line segment,

In the equation of transformation,

Therefore,

This equation is satisfied only if