Keeping the bead fixed while moving the trap center shows that force is sourced by displacement. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
A far trap center makes a six-metre displacement
Hold bead position at 6 metres and stiffness at 2 newtons per metre. Center 0 gives displacement 6 metres and restoring force 12 newtons.
2⋅(6−0)=12
Moving the center closer shortens the vector
The bead still sits at 6 metres. Moving the trap center to 2 metres leaves displacement 4 metres, so the same stiffness gives 8 newtons.
2⋅(6−2)=8
Three center rows expose the displacement source
Center 4 metres leaves only 2 metres of displacement. The restoring force falls to 4 newtons because the vector, not the bead coordinate alone, is the source.