Final answer:
To find the distance from point P to a plane, use the vector cross and dot products to derive the perpendicular height from P to the plane. Distances between points remain invariant under coordinate system rotations due to the Pythagorean theorem, which is orientation-independent.
Step-by-step explanation:
To find the distance d from point P to the plane defined by points Q, R, and S, we can use the vector formula d = |(a × b) · c| / |a × b| where a = QR, b = QS, and c = CP. The cross product a × b gives a vector perpendicular to the plane, and taking the dot product with c gives the component of c in the direction of a × b (which is perpendicular to the plane), thus giving the height of the parallelepiped defined by a, b, and c. Dividing by the magnitude of a × b normalizes this height to give the distance from point P to the plane.
Regarding the invariance of distances under rotation, when we rotate a coordinate system, the actual distances between points do not change. For instance, the distance between points P and Q remains constant, as does the distance from P to the origin. This invariance can be shown mathematically because the distance formula (whether between two points or from a point to the origin) is based on the Pythagorean theorem, which is independent of the orientation of the coordinate system. For point P with coordinates (x, y, z), the distance to the origin r is given by r² = x² + y² + z², which remains the same under rotation.