Thornton & Marion, Classical Dynamics, Fifth 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 orthogornal.
We should know about
1) Transformation (rotation) matrix 2) Orthogonality condition
We assume a point is represented in the coordinate system by , and it can be also represented in the coordinate system by . These coordinate system have the same origin. The length of the line segment from the origin and the point is
Since the transformation matrix preserves the length of the line segment, thus, In the equation of transformation, using the fact that the index is a dummy variable, so we can write the above equation like this: Therefore, This equation is satisfied only if