# 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.

1. Transformation (rotation) matrix
2. 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

$latex l^2 = x^2_1 + x^2_2 + x^2_3 = \\sum_{i=1}^{3} x^2_i$
$latex l\’^2 = x\’^2_1 + x\’^2_2 + x\’^2_3 = \\sum^3_{i=1} x\’^2_i$

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

$latex l^2 = l\’^2$
thus,

$latex \\sum^3_{i=1} x^2_i = \\sum^3_{i=1} x\’^2_i$

In the equation of transformation,

$latex x\’_i = \\sum_j \\lambda_{ij} x_j$
using the fact that the index $latex j$ is a dummy variable, so we can write the above equation like this:

$latex \\sum^3_{i=1} x\’^2_i = \\sum^3_{i=1} ( \\sum^3_{j=1} \\lambda_{ij} x_j) ( \\sum^3_{k=1} \\lambda_{ik} x_k)$

Therefore,

$latex \\sum^3_{i} x^2_i = \\sum^3_{i=1} ( \\sum^3_{j=1} \\lambda_{ij} x_j) ( \\sum^3_{k=1} \\lambda_{ik} x_k) = \\sum_{j,k}^3 x_j x_k (\\sum_i^3 \\lambda_{ij} \\lambda_{ik})$

This equation is satisfied only if

$latex \\sum_i \\lambda_{ij}\\lambda_{ik} = \\delta_{jk}$