12. Mathematical Skills Required to Fully Understand SEGMENT-Landslide

In this book, we suppose the reader has basic vector concept and for the convenience, listed the basic vector operations in the Appendix D. In the following discussion, we stick to the right-hand system. Suppose the unit directional vectors of a Cartesian system is \( {\widehat{e}}_1,\kern0.5em {\widehat{e}}_2\ \mathrm{and}\ {\widehat{e}}_3 \). Position vector \( \overrightarrow{x} \) (of a granular element), can be decomposed into the \( {\widehat{e}}_1,\kern0.5em {\widehat{e}}_2\ \mathrm{and}\ {\widehat{e}}_3 \) directions, so that \( \overrightarrow{x}={x}_1{\widehat{e}}_1+{x}_2{\widehat{e}}_2+{x}_3{\widehat{e}}_3 \). Now introduce a new Cartesian coordinate system, it is obtained by rotation of the original one. Unit vectors in the rotated system called \( {{\widehat{e}}_1}^{\hbox{'}},\kern0.5em {{\widehat{e}}_2}^{\hbox{'}}\ \mathrm{and}\ {{\widehat{e}}_3}^{\hbox{'}}. \) Then the same vector can be write in the new (rotated system) as: \( \overrightarrow{x}={x}_1^{\hbox{'}}{\widehat{e}}_1^{\hbox{'}}+{x}_2^{\hbox{'}}{\widehat{e}}_2^{\hbox{'}}+{x}_3^{\hbox{'}}{\widehat{e}}_3^{\hbox{'}}={\displaystyle \sum_{i=1}^3{x}_i^{\hbox{'}}{\widehat{e}}_i^{\hbox{'}}} \). Generally speaking, x _i  ≠ x _i ^' , i = 1,2,3. A natural question is how are the components of \( \overrightarrow{x} \) in the new system related to its components in the original one? Take x ₂ ^' for an example, \( {x}_2^{\hbox{'}}=\overrightarrow{x}\bullet {\widehat{e}}_2^{\hbox{'}}={\displaystyle \sum_{i=1}^3{x}_i{\widehat{e}}_i}\cdot {\widehat{e}}_2^{\hbox{'}}={\displaystyle \sum_{i=1}^3{x}_i\left|{\widehat{e}}_i\right|}\left|{\widehat{e}}_2^{\hbox{'}}\right| \cos \left({\widehat{e}}_i,{\widehat{e}}_2^{\hbox{'}}\right)={\displaystyle \sum_{i=1}^3{x}_i}{C}_{i,2} \)