Final answer:
Transformation 1A is a translation to the right by 1 unit on the coordinate plane, moving points horizontally to the right. Additionally, the distance of a point from the origin remains invariant under coordinate system rotations, as shown by the equation x'^2 + y'^2 = x^2 + y^2.
Step-by-step explanation:
The question involves transformations on the coordinate plane. Transformation 1A refers to a translation to the right by 1 unit, which can be visualized as moving each point on the plane horizontally to the right side of the coordinate system by one unit. This transformation preserves the shape and size of figures but changes their position.
To demonstrate the invariance of distance from the origin under rotations, consider point P with coordinates (x, y). A rotation of the coordinate system does not change the actual distance of a point from the origin, since the rotation is a rigid motion. If the coordinate system is rotated by any angle, the new coordinates (x', y') will still satisfy the equation x'^2 + y'^2 = x^2 + y^2, showing that the distance remains the same.
In terms of vectors and directionality, rotating a vector counterclockwise or clockwise, increasing its magnitude through dilation, or changing its direction with reflection are all transformations that can be applied in a coordinate system.