Thornton & Marion (5th Edition), Chapter 01, Problem 03 Solution

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.

You should know about

  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

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