The robot's reachable positions on the x-y plane can be determined by vector addition, combining its basic movements in various sequences. The positions are combinations of the vectors (1, 3), (-1, -3), (3, 7), and (-3, -7) that result in a net position on the plane.
We are tasked with finding out which locations on the x-y plane a simple robot can reach given its movement capabilities. The robot has four types of directional movements it can make from the origin (0,0), each being a vector displacement. These vectors are: (1, 3), (-1, -3), (3, 7), and (-3, -7).
To determine which points the robot can reach, we can use vector addition. The robot's movements can be represented as adding these vectors to its current location. For example, moving right 1 inch and up 3 inches from the origin would result in the position (1, 3). If the robot then moves left 1 inch and down 3 inches, it will have returned to the origin (0, 0).
The key to this problem is recognizing that the points the robot can reach will depend on combinations of these vectors. By repeatedly adding or subtracting these vectors from the current position, the robot can reach a new position on the x-y plane. The robot's reachable positions will be any combination of the vector moves that maintain a net position on the x-y plane. It can be observed that smaller vectors like (1, 3) can be composed by subtracting the larger vector (3, 7) from smaller multiples of itself, indicating that there's an underlying relationship between these vectors.