# Displacement (vector)

A **displacement** is the shortest distance from the initial to the final position of a point P.^{[1]}
Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P. A "displacement vector" represents the length and direction of that imaginary straight path.

A position vector expresses the position of a point P in space in terms of a displacement from an arbitrary reference point O (typically the origin of a coordinate system). Namely, it indicates both the distance and direction of an imaginary motion along a straight line from the reference position to the actual position of the point.

A displacement may be also described as a 'relative position': the final position of a point (* R_{f}*) relative to its initial position (

*), and a displacement vector can be mathematically defined as the difference between the final and initial position vectors:*

**R**_{i}In considering motions of objects over time the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The velocity then is distinct from the instantaneous speed which is the time rate of change of the distance traveled along a specific path. The velocity may be equivalently defined as the time rate of change of the position vector. If one considers a moving initial position, or equivalently a moving origin (e.g. an initial position or origin which is fixed to a train wagon, which in turn moves with respect to its rail track), the velocity of P (e.g. a point representing the position of a passenger walking on the train) may be referred to as a relative velocity, as opposed to an absolute velocity, which is computed with respect to a point which is considered to be 'fixed in space' (such as, for instance, a point fixed on the floor of the train station).

For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity. (Note that the average velocity, as a vector, differs from the average speed that is the ratio of the path length — a scalar — and the time interval.)

## Contents

## Rigid body

In dealing with the motion of a rigid body, the term *displacement* may also include the rotations of the body. In this case, the displacement of a particle of the body is called **linear displacement** (displacement along a line), while the rotation of the body is called **angular displacement**.

## Derivatives

For a position vector * s* that is a function of time

*t*, the derivatives can be computed with respect to

*t*. These derivatives have common utility in the study of kinematics, control theory, and other sciences and engineering disciplines.

- (where d
is an infinitesimally small displacement)**s**

These common names correspond to terminology used in basic kinematics.^{[2]} By extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering and physics.

## See also

## References

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