Point \(P~=~(3,~2)\) partitions the directed segment from \(A~=~(0,~0)\) to \(B~=~(9,~6)\). Express \(P\) as a common fraction of the displacement from \(A\) to \(B\), then give the internal ratio \(AP:PB\) in simplest form.

To recover a partition ratio, compare the two pieces in their requested order rather than comparing one piece with the whole. We’ll compute the start-to-point and point-to-end displacement vectors, confirm their components share one scale factor, reduce those proportional sizes, and check that the point lies internally on the segment.