Position (geometry)

 ( Kinematic Properties ) 

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O, and its direction represents the angular orientation with respect to given reference axes. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P. In other words, it is the displacement or translation that maps the origin to P:The term displacement is mainly used in mechanics, while translation is used in geometry.

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

The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to and of any dimension.Keller, F. J., Gettys, W. E. et al. (1993), p. 28–29.

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Relative position The relative position of a point Q with respect to point P is the Euclidean vector resulting from the subtraction of the two absolute position vectors (each with respect to the origin):

$\backslash Delta\; \backslash mathbf\{r\}=\backslash mathbf\{s\}\; -\; \backslash mathbf\{r\}=\backslash overrightarrow\{PQ\},$

where $\backslash mathbf\{s\}=\backslash overrightarrow\{OQ\}$. The relative direction between two points is their relative position normalized as a unit vector.

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Definition and representation

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Three dimensions In three dimensions, any set of three-dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used.

Commonly, one uses the familiar Cartesian coordinate system, or sometimes spherical polar coordinates, or cylindrical coordinates:

$\backslash begin\{align\}$

\mathbf{r}(t)
& \equiv \mathbf{r}(x,y,z) \equiv x(t)\mathbf{\hat{e}}_x + y(t)\mathbf{\hat{e}}_y + z(t)\mathbf{\hat{e}}_z  \\
& \equiv \mathbf{r}(r,\theta,\phi) \equiv r(t)\mathbf{\hat{e}}_r\big(\theta(t), \phi(t)\big) \\
& \equiv \mathbf{r}(r,\phi,z) \equiv r(t)\mathbf{\hat{e}}_r\big(\phi(t)\big) + z(t)\mathbf{\hat{e}}_z, \\
     

\end{align}

where t is a parameter, owing to their rectangular or circular symmetry. These different coordinates and corresponding basis vectors represent the same position vector. More general curvilinear coordinates could be used instead and are in contexts like continuum mechanics and general relativity (in the latter case one needs an additional time coordinate).

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n dimensions Linear algebra allows for the abstraction of an n-dimensional position vector. A position vector can be expressed as a linear combination of basis vectors:

(2025). 9780521861533, Cambridge University Press. . ISBN 9780521861533

(2025). 9780071543521, McGraw Hill. . ISBN 9780071543521

$\backslash mathbf\{r\}\; =\; \backslash sum\_\{i=1\}^n\; x\_i\; \backslash mathbf\{e\}\_i\; =\; x\_1\; \backslash mathbf\{e\}\_1\; +\; x\_2\; \backslash mathbf\{e\}\_2\; +\; \backslash dotsb\; +\; x\_n\; \backslash mathbf\{e\}\_n.$

The set of all position vectors forms position space (a vector space whose elements are the position vectors), since positions can be added (vector addition) and scaled in length (scalar multiplication) to obtain another position vector in the space. The notion of "space" is intuitive, since each x_i ( i = 1, 2, …, n) can have any value, the collection of values defines a point in space.

The dimension of the position space is n (also denoted dim( R) = n). The coordinates of the vector r with respect to the basis vectors e _i are x _i. The vector of coordinates forms the coordinate vector or n-tuple ( x₁, x₂, …, x_n).

Each coordinate x_i may be parameterized a number of t. One parameter x_i( t) would describe a curved 1D path, two parameters x_i( t₁, t₂) describes a curved 2D surface, three x_i( t₁, t₂, t₃) describes a curved 3D volume of space, and so on.

The linear span of a basis set B = { e₁, e₂, …, e _n} equals the position space R, denoted span( B) = R.

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Applications

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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.

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Mechanics In any equation of motion, the position vector r( t) is usually the most sought-after quantity because this function defines the motion of a particle (i.e. a point mass) – its location relative to a given coordinate system at some time t.

To define motion in terms of position, each coordinate may be parametrized by time; since each successive value of time corresponds to a sequence of successive spatial locations given by the coordinates, the continuum limit of many successive locations is a path the particle traces.

In the case of one dimension, the position has only one component, so it effectively degenerates to a scalar coordinate. It could be, say, a vector in the x direction, or the radial r direction. Equivalent notations include

$\backslash mathbf\{x\}\; \backslash equiv\; x\; \backslash equiv\; x(t),\; \backslash quad\; r\; \backslash equiv\; r(t),\; \backslash quad\; s\; \backslash equiv\; s(t).$

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Derivatives For a position vector r that is a function of time t, the can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, engineering and other sciences.

Velocity

$\backslash mathbf\{v\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{r\}\}\{\backslash mathrm\{d\}t\},$

where d r is an infinitesimally small displacement (vector).

Acceleration

$\backslash mathbf\{a\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{v\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash mathrm\{d\}^2\backslash mathbf\{r\}\}\{\backslash mathrm\{d\}t^2\}.$

Jerk

$\backslash mathbf\{j\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{a\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash mathrm\{d\}^2\backslash mathbf\{v\}\}\{\backslash mathrm\{d\}t^2\}\; =\; \backslash frac\{\backslash mathrm\{d\}^3\backslash mathbf\{r\}\}\{\backslash mathrm\{d\}t^3\}.$

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 Taylor series, enabling several analytical techniques in engineering and physics.

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See also

• Affine space
• Coordinate system
• Horizontal position
• Line element
• Parametric surface
• Position fixing
• Position four-vector
• Six degrees of freedom
• Vertical position

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Notes

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

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External links

Categories: Kinematic Properties

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