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

Chapter 01. Matrices, Vectors, and Vector Calculus

Problem 01. The transformation matrix

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

\lambda_{11} = \cos(x'_1,x_1) = \frac{\sqrt{2}}{2}
\lambda_{12} = \cos(x'_1,x_2) = 0
\lambda_{13} = \cos(x'_1,x_3) = \cos 135^{\circ} = - \frac{\sqrt{2}}{2} 
\lambda_{21} = \cos(x'_2,x_1) = 0
\lambda_{22} = \cos(x'_2,x_2) = 1
\lambda_{23} = \cos(x'_2,x_3) = 0
\lambda_{31} = \cos(x'_3,x_1) = \frac{\sqrt{2}}{2}
\lambda_{32} = \cos(x'_3,x_2) = 0
\lambda_{33} = \cos(x'_3,x_3) = \frac{\sqrt{2}}{2}


Therefore, the transformation matrix is

\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}

 

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