Position and Displacement Vectors

To describe the location of a particle moving in a plane, first choose a fixed reference

point $O$, called the origin, and a pair of mutually perpendicular axes through it (the

x-axis and the y-axis). The position vector of the particle at any instant is the vector

$\vec r$ drawn from the origin $O$ to the particle's location $P$ at that instant. In

component form, if $P$ has coordinates $(x, y)$,

where $\hat i$ and $\hat j$ are unit vectors along the x- and y-axes.

As the particle moves, its position vector changes with time. If the particle is at position

vector $\vec r_1$ at time $t_1$ and at position vector $\vec r_2$ at a later time $t_2$, its

displacement over that interval is the vector

Displacement is drawn as the straight-line arrow from the particle's initial position to its

final position — it depends only on where the particle started and ended, never on the

actual path taken in between.

It is important not to confuse displacement with distance. Distance is the total length

of the actual path traversed and is a scalar; displacement is the straight-line vector from

start to end. The magnitude of the displacement can never exceed the distance travelled, and

the two are equal only for motion that proceeds in a single, unchanging direction (as in

straight-line motion covered in the previous sub-topic).

One subtlety worth noting: the position vector $\vec r$ of a point depends on the choice of …