Displacement is the straight-line change in position from a starting point to an ending point — captured as a single vector that carries both how far and in which direction. Unlike distance travelled (which counts every wiggle of the path), displacement cares only about where you began and where you ended.
From Two Points
Suppose a particle moves from point A to point B. Its displacement is the vector AB drawn from A (tail) to B (head). If A and B have position vectors a and b (measured from the origin), then travelling from the origin to A and then along the displacement must land you at B: a+AB=b. Rearranging:
AB=b−a(position vector of head−position vector of tail)
A handy memory aid: "head minus tail."
Its Magnitude Is the Distance
The length of the displacement vector is the straight-line distance between the two points:
∣AB∣=∣b−a∣.
In coordinates, if A=(x1,y1,z1) and B=(x2,y2,z2), then
AB=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^,
and its magnitude is the familiar distance formula (x2−x1)2+(y2−y1)2+(z2−z1)2.
Displacement vs. Distance
Note
Distance travelled is a scalar that depends on the whole route; displacement is a vector that depends only on the endpoints. Walk a full loop and return home: the distance is large, but the displacement is the zero vector.