Final answer:
The provided question is fragmented and lacks clarity. To provide an accurate physics explanation regarding the invariance of distance under coordinate rotation, additional context would be necessary. However, the invariance can be related to the Pythagorean theorem and coordinate transformation.
Step-by-step explanation:
The question provided seems to contain multiple parts, and it appears to be from a physics context. However, there is critical information missing that doesn't allow for a complete understanding of the initial problem. To help the student, more specific details about the original question would be necessary to give an accurate answer, especially since parts of the question text seem unrelated or improperly formatted.
In physics, the invariance of the distance between points under rotation is an essential concept that stems from the idea that physical laws and measurements should remain consistent regardless of the orientation of the coordinate system. This invariance can be proven using Pythagorean theorem or vector transformation in coordinate systems. For the case of the distance of a point P to the origin being invariant under rotation, we reference that the distance is calculated as the square root of the sum of squares of the coordinates (e.g., √(x² + y²) in 2D space), which does not change under rotation.