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

Thornton & Marion, Classical Dynamics, 5th Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 03. Transformation Matrix

The problem asks you to

  • find a transformation matrix that satisfying 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 \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

\vec{e_1}' = \vec{e_2}
\vec{e_2}' = \vec{e_3}
\vec{e_3}' = \vec{e_1}
 

So, the transformation matrix is

\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)
\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,

\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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