Position (vector)

In geometry, a position, location, or radius vector, usually denoted $\mathbf{r}$, is a vector which represents the position of a point P in space in relation to an arbitrary reference origin O. It corresponds to the displacement from O to P:

$\mathbf{r} = \overrightarrow{OP}.$

The concept typically applies to two- or three-dimensional space, but can be easily generalized to Euclidean spaces with a higher number of dimensions.

Applications

• In linear algebra, a position vector can be expressed as a linear combination of basis vectors.

• The kinematic movement of a point mass can be described by a vector-valued function giving the position $\mathbf{r}(t)$ as a function of the scalar time parameter t. These are used in mechanics and dynamics to keep track of the positions of particles, point masses, or rigid objects.

• In differential geometry, position vector fields are used to describe continuous and differentiable space curves, in which case the independent parameter needs not be time, but can be (e.g.) arc length of the curve.

Derivatives of Position

For a position vector $\vec r$ 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, engineering and other sciences.

Velocity

$\vec v =\frac {d \vec r} {dt}$ (where $d \vec r$ is an infinitesimally small displacement (vector))

Acceleration

$\vec a =\frac {d \vec v} {dt}=\frac {d ^2\vec r} {dt^2}$

Jerk

$\vec j =\frac {d \vec a} {dt}=\frac {d ^2\vec v} {dt^2}=\frac {d ^3\vec r} {dt^3}$

These names for the first, second and third derivative of position are commonly used in basic kinematics. 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.

Relationship to displacement vectors

A displacement vector can be defined as the action of uniformly translating spatial points in a given direction over a given distance. Thus the addition of displacement vectors expresses the composition of these displacement actions and scalar multiplication as scaling of the distance. With this in mind we may then define a position vector of a point in space as the displacement vector mapping a given origin to that point. Note thus position vectors depend on a choice of origin for the space, as well as displacement vectors depend on the choice of an initial point.

See also

• Affine space
• Curve
• Parametric surface
• Displacement vector

References

1. Keller, F. J, Gettys, W. E. et al. (1993). "Physics: Classical and modern" 2nd ed. McGraw Hill Publishing