Position and Displacement

One general way of locating a particle (or particle-like object) is with a position vector, which is a vector that extends from a reference point (usually the origin of a coordinate system) to the particle. In the unit-vector notation, can be written

, (4-1)

where , , and are the vector components of , and the coefficients , , and are its scalar components.

The coefficients , , and give the particle's location along the coordinate axes and relative to the origin; that is, the particle has the rectangular coordinates ( , , ). For instance, Fig. 4-1 shows a particle with position vector

(4-2)

Fig. 4-1

and rectangular coordinates . Along the axis the particle is 3 m from the origin, in the direction. Along the axis it is 2 m from the origin, in the direction. Along the axis it is 5 m from the origin, in the direction.

As a particle moves, its position vector changes in such a way that the vector always extends to the particle from the reference point (the origin). If the position vector changes, say, from to during a certain time interval, then the particle's displacementduring that time interval is

(4-2)

Using the unit-vector notation, we can rewrite this displacement as

or as

(4-3)

where coordinates ( , , ) correspond to position vector and coordinates ( , , )correspond to position vector . We can also rewrite the displacement by substituting for , for , and for :

(4-3)

Example 4-1

In Fig. 4-2, the position vector for a particle is initially

Fig. 4-2

and then later is

.

What is the particle's displacement from to ?

Solution. The Key Ideais that the displacement is obtained by subtracting the initial position vector from the later position vector . That is most easily done by components:

(Answer)

This displacement vector is parallel to the plane, because it lacks any component, a fact that is easier to see in the numerical result than in Fig. 4-2.