Thornton & Marion (Fifth Edition), Chapter 01, Exercise Problem 03 Solution

Chapter 01. Matrices, Vectors, and Vector Calculus

Problem 03. Transformation Matrix

The problem asks you to
find a transformation matrix that satisfying some condition.
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)$$\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