# 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 isSince 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