# Thornton & Marion, Classical Dynamics of Particles and Systems, 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 $latex P$ is represented in the $latex x_i$ coordinate system by $latex P(x_1, x_2, x_3)$, and it can be also represented in the $latex x\’_i$ coordinate system by $latex P(x\’_1, x\’_2, x\’_3)$.

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

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

In the equation of transformation,

Therefore,

This equation is satisfied only if

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