A position vector locates a point relative to the origin, while a displacement vector describes the change in position from one point to another. Both have magnitude and direction, but they use different starting points.
Let be the origin and let and be two points.
| Vector type | Meaning | Calculation |
|---|---|---|
| Position vector | Runs from the origin to a point |
| Displacement vector | Runs from one point to another |
Position vectors depend on the chosen origin: moving the origin changes and . However, the displacement between the same two points is unchanged because the same origin shift is subtracted from both position vectors. This is why displacement represents relative movement rather than an absolute location in any chosen coordinate system.
The subtraction order matters. To find the displacement from to , subtract the position vector of the starting point from that of the finishing point:
For example, suppose
Then
This means moving from to requires a change of units in the positive -direction, units in the positive -direction, and units in the negative -direction.
Reversing the journey reverses the vector: . Its magnitude is still the same, so as distances, even though the two displacement vectors have opposite directions. Also, a displacement vector can be translated parallel to itself without changing its components; unlike a position vector, it does not need to begin at the origin.
The distance between the points is the magnitude of this displacement:
A common misconception is that the coordinates of give the displacement from . They give 's position relative to the origin; displacement requires .
This is Mathematics: Analysis and Approaches HL, sub-topic AHL 3.12: Vector definitions, and may appear on Paper 1 or Paper 2. Show the subtraction in the correct order before calculating components, and distinguish the vector from its scalar magnitude .
