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

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

### Problem 01. The transformation matrix

#### Solution

The direction cosines $latex \\lambda_{ij}$ can be determined using the definition of Equation (1.3).

$latex \\lambda_{11} = \\cos(x\’_1,x_1) = \\frac{\\sqrt{2}}{2}$

$latex \\lambda_{12} = \\cos(x\’_1,x_2) = 0$

$latex \\lambda_{13} = \\cos(x\’_1,x_3) = \\cos 135^{\\circ} = – \\frac{\\sqrt{2}}{2}$

$latex \\lambda_{21} = \\cos(x\’_2,x_1) = 0$

$latex \\lambda_{22} = \\cos(x\’_2,x_2) = 1$

$latex \\lambda_{23} = \\cos(x\’_2,x_3) = 0$

$latex \\lambda_{31} = \\cos(x\’_3,x_1) = \\frac{\\sqrt{2}}{2}$

$latex \\lambda_{32} = \\cos(x\’_3,x_2) = 0$

$latex \\lambda_{33} = \\cos(x\’_3,x_3) = \\frac{\\sqrt{2}}{2}$

Therefore, the transformation matrix is $latex \\lambda = \\begin{pmatrix} \\frac{\\sqrt{2}}{2} & 0 & -\\frac{\\sqrt{2}}{2} \\\\ 0 & 1 & 0 \\\\ \\frac{\\sqrt{2}}{2} & 0 & \\frac{\\sqrt{2}}{2} \\end{pmatrix}$