📢Impact of Non-Orthogonal Coordinates on Geometric Measurement
This problem beautifully illustrates how a non-orthogonal coordinate system impacts fundamental geometric measurements. The most important result is the non-zero off-diagonal term in the metric tensor, which is the defining characteristic of a non-orthogonal system, confirming that the new basis vectors are not perpendicular. Furthermore, the diagonal element shows the basis vector is not normalized. This non-trivial metric structure means that the formula for the length of a curve must include a cross-term, which accounts for the angle between the axes. If the system were Cartesian, this term would vanish, simplifying the line element back to the standard Pythagorean formula.
