Final answer:
The position vector PQ for points P(1, 3) and Q(−1, 4) is calculated by subtracting P from Q, resulting in -2î + 1Ƶ. The distance between the points, as determined by the magnitude of this vector, is invariant under coordinate system rotations.
Step-by-step explanation:
To find the position vector PQ, you subtract the coordinates of point P from those of point Q. For points P(1, 3) and Q(−1, 4), the position vector PQ can be calculated as Q - P, which gives us (-1 - 1) î + (4 - 3) Ƶ. Simplifying this, we get the position vector PQ = -2î + 1Ƶ.
In reference to demonstrations of invariance under rotations, we consider that the distance between two points is given by the magnitude of the displacement vector between them. This distance is calculated using the Pythagorean theorem in Euclidean space, and it remains constant regardless of the orientation of the coordinate system, which confirms its invariance under rotations.