Thornton & Marion, Classical Dynamics of Particles and Systems, 5th Edition
Chapter 1. Matrices, Vectors, and Vector Calculus
Problem 06. Another proof of the orthogonality condition
The problem asks you to
- prove that Equation (1.15) can be obtained by using the fact that the transformation matrix preserves the length of the line segment.
This problem assumes
- that the coordinate systems are both orthogonal.
We should know about
- Transformation (rotation) matrix
- Orthogonality condition
Solution
We assume a point $latex P$ is represented in the $latex x_i$ coordinate system by $latex P(x_1, x_2, x_3)$, and it can be also represented in the $latex x\’_i$ coordinate system by $latex P(x\’_1, x\’_2, x\’_3)$.
These coordinate systems have the same origin. The length of the line segment from the origin and the point $latex P$ is
Since the transformation matrix preserves the length of the line segment,
In the equation of transformation,
Therefore,
This equation is satisfied only if
Leave a Reply