Which three organ systems are directly interacting when a persons brain sends a signal to the persons arm, causing the arm to move up and down

 The three organ systems involved in arm movements are:  
 1. Nervous system (central and peripheral)
The movements of the muscles are coordinated and controlled by the brain. The motor cortex of the brain sends an electrical signal through the spinal cord and peripheral nerves to the muscles which cause them to contract and make a move.
 2. Muscle system
The skeletal muscles are attached to arm bone and via quickly and powerfully contractions (shorten or tighten) make a movement.
 3. Skeletal system
Joints are part of the skeletal system and are the connections made between bones. They are highly moveable and that makes the skeleton flexible and capable for movement.

No you are not wrong