# Thornton & Marion, Classical Dynamics of Particles and Systems, 5th Edition

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

### Problem 03. Transformation Matrix

#### The problem asks you to

• find a transformation matrix that satisfies some conditions.

#### This problem gives (or assumes)

• This matrix rotates a rectangular coordinate through an angle of 120 degrees about an axis making equal angles with the original one.

1. Direction cosine $latex \\lambda_{ij} = cos(x\’_i, x_j)$
2. Transformation matrix

#### Solution

We can see the relation between the rotated and the original coordinates system. This picture shows that

$latex \\vec{e_1}\’ = \\vec{e_2}$
$latex \\vec{e_2}\’ = \\vec{e_3}$
$latex \\vec{e_3}\’ = \\vec{e_1}$

So, the transformation matrix is

$latex \\lambda = \\left( \\begin{array}{ccc} \\lambda_{11} & \\lambda_{12} & \\lambda_{13} \\\\ \\lambda_{21} & \\lambda_{22} & \\lambda_{23} \\\\ \\lambda_{31} & \\lambda_{32} & \\lambda_{33} \\\\ \\end{array} \\right)$
$latex \\lambda = \\left( \\begin{array}{ccc} \\cos 90^\\circ & \\cos 0^\\circ & \\cos 90^\\circ \\\\ \\cos 90^\\circ & \\cos 90^\\circ & \\cos 0^\\circ \\\\ \\cos 0^\\circ & \\cos 90^\\circ & \\cos 90^\\circ \\\\ \\end{array} \\right)$

Therefore,

$latex \\lambda = \\left( \\begin{array}{ccc} 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ \\end{array} \\right)$

#### Reference

https://math.stackexchange.com/questions/1599561/determining-the-transformation-matrix-r?newreg=f85754c5968d4b7fae383aabe7bfd2a5