Thornton & Marion, Classical Dynamics, Fifth 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 orthogornal.

We should know about

1) Transformation (rotation) matrix
2) 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 system 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,
  
thus,
  
In the equation of transformation,
  
using the fact that the index  is a dummy variable, so we can write the above equation like this:
  
Therefore,
  
This equation is satisfied only if