# Position vector

Two points and their position vectors
Position vectors (here denoted by and ) in the Cartesian coordinate system${\ displaystyle {\ vec {r}} _ {P}}$${\ displaystyle {\ vec {r}} _ {Q}}$

In mathematics and in physics, a position vector (also radius vector or position vector ) of a point is a vector that points from a fixed reference point to this point (location). In elementary and synthetic geometry , these vectors can be defined as classes of arrows with the same displacement or as parallel displacements .

Position vectors make it possible to use vector calculation for the description of points , sets of points and images . If a Cartesian coordinate system is used as a basis, the origin of the coordinates is generally chosen as the reference point for the position vectors of the points. In this case the coordinates of a point with respect to this coordinate system coincide with the coordinates of its position vector.

In analytical geometry , position vectors are used to describe images of an affine or Euclidean space and to describe sets of points (such as straight lines and planes ) by means of equations and parametric representations .

In physics , position vectors are used to describe the location of a body in Euclidean space. In coordinate transformations, position vectors show a different transformation behavior than covariant vectors .

## Spellings

In geometry, the reference point (origin) is usually referred to as (for Latin origo ). The notation for the position vector of a point is then: ${\ displaystyle O}$${\ displaystyle P}$

${\ displaystyle {\ overrightarrow {OP}}}$

Occasionally the lowercase letters with vector arrows are also used, which correspond to the uppercase letters with which the points are designated, for example:

${\ displaystyle {\ vec {p}} = {\ overrightarrow {OP}}, \ {\ vec {q}} = {\ overrightarrow {OQ}}, \ {\ vec {a}} = {\ overrightarrow {OA }}, \ {\ vec {b}} = {\ overrightarrow {OB}}, \ \ dots, \ {\ vec {x}} = {\ overrightarrow {OX}}}$

The notation that the capital letter denoting the point is provided with a vector arrow is also common:

${\ displaystyle {\ vec {P}} = {\ overrightarrow {OP}}, \ {\ vec {Q}} = {\ overrightarrow {OQ}}, \ {\ vec {A}} = {\ overrightarrow {OA }}, \ {\ vec {B}} = {\ overrightarrow {OB}}, \ \ dots, \ {\ vec {X}} = {\ overrightarrow {OX}}}$

In physics in particular, the position vector is also called the radius vector and is written with a vector arrow as or (especially in theoretical physics) in semi-bold as . ${\ displaystyle {\ vec {r}}}$${\ displaystyle \ mathbf {r}}$

## Examples and applications in geometry

### Connection vector

For the connection vector of two points and with the position vectors and the following applies: ${\ displaystyle {\ overrightarrow {PQ}}}$${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle {\ vec {p}} = {\ overrightarrow {OP}}}$${\ displaystyle {\ vec {q}} = {\ overrightarrow {OQ}}}$

${\ displaystyle {\ overrightarrow {PQ}} = {\ overrightarrow {OQ}} - {\ overrightarrow {OP}} = {\ vec {q}} - {\ vec {p}}}$

### Cartesian coordinates

The following applies to the coordinates of the position vector of the point with the coordinates : ${\ displaystyle {\ overrightarrow {OP}}}$${\ displaystyle P}$${\ displaystyle (p_ {1}, p_ {2}, p_ {3})}$

${\ displaystyle {\ overrightarrow {OP}} = {\ begin {pmatrix} p_ {1} \\ p_ {2} \\ p_ {3} \ end {pmatrix}}}$

### shift

A shift around the vector maps the point onto the point . Then the following applies to the position vectors: ${\ displaystyle {\ vec {v}}}$${\ displaystyle X}$${\ displaystyle X ^ {\ prime}}$

${\ displaystyle {\ overrightarrow {OX '}} = {\ overrightarrow {OX}} + {\ vec {v}}}$
${\ displaystyle {\ vec {x}} '= {\ vec {x}} + {\ vec {v}}}$

### Rotation around the origin

A rotation in the plane with the center of rotation to the angle counter -clockwise , with the aid of a as follows in Cartesian coordinate rotation matrix will be described: If the position vector of a point and the position vector of the pixel , then: ${\ displaystyle O}$ ${\ displaystyle \ varphi}$${\ displaystyle {\ vec {x}} = {\ tbinom {x_ {1}} {x_ {2}}} = {\ overrightarrow {OX}}}$${\ displaystyle X}$${\ displaystyle {\ vec {x}} '= {\ tbinom {x_ {1}'} {x_ {2} '}} = {\ overrightarrow {OX'}}}$${\ displaystyle X '}$

${\ displaystyle {\ begin {pmatrix} x_ {1} '\\ x_ {2}' \ end {pmatrix}} = {\ begin {pmatrix} \ cos \ varphi & - \ sin \ varphi \\\ sin \ varphi & \ cos \ varphi \ end {pmatrix}} {\ begin {pmatrix} x_ {1} \\ x_ {2} \ end {pmatrix}}}$

### Affine figure

A general affine mapping that maps the point to the point can be represented with position vectors as follows: ${\ displaystyle X}$${\ displaystyle X '}$

${\ displaystyle {\ vec {x}} '= L ({\ vec {x}}) + {\ vec {v}}}$

Here, the position vector von , the position vector von , is a linear mapping and a vector that describes a displacement. In Cartesian coordinates, the linear mapping can be represented by a matrix and the following applies: ${\ displaystyle {\ vec {x}}}$${\ displaystyle X}$${\ displaystyle {\ vec {x}} '}$${\ displaystyle X '}$${\ displaystyle L}$${\ displaystyle {\ vec {v}}}$${\ displaystyle L}$${\ displaystyle A}$

${\ displaystyle {\ vec {x}} = A \ cdot {\ vec {x}} + {\ vec {v}}}$

In three-dimensional space this results in:

${\ displaystyle {\ begin {pmatrix} x_ {1} '\\ x_ {2}' \\ x_ {3} '\ end {pmatrix}} = {\ begin {pmatrix} a_ {11} & a_ {12} & a_ {13} \\ a_ {21} & a_ {22} & a_ {23} \\ a_ {31} & a_ {32} & a_ {33} \ end {pmatrix}} \, {\ begin {pmatrix} x_ {1} \ \ x_ {2} \\ x_ {3} \ end {pmatrix}} + {\ begin {pmatrix} v_ {1} \\ v_ {2} \\ v_ {3} \ end {pmatrix}}}$

Corresponding representations are also available for other dimensions.

### Parametric representation of a straight line

The straight line through the points and contains exactly those points whose position vector is the representation ${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle X}$${\ displaystyle {\ vec {x}}}$

${\ displaystyle {\ vec {x}} = {\ overrightarrow {OP}} + t \, {\ overrightarrow {PQ}}}$ With ${\ displaystyle t \ in \ mathbb {R}}$

owns. One speaks here of the parametric form of a straight line equation .

### Normal form of the plane equation

The plane through the point (support point) with normal vector contains exactly those points whose position vector corresponds to the normal equation${\ displaystyle P}$ ${\ displaystyle {\ vec {n}}}$${\ displaystyle X}$${\ displaystyle {\ vec {x}}}$

${\ displaystyle {\ vec {x}} \ cdot {\ vec {n}} = {\ vec {p}} \ cdot {\ vec {n}}}$

Fulfills. It is the position vector ( support vector ) of the support point and the Malpunkt denotes the scalar product . ${\ displaystyle {\ vec {p}}}$${\ displaystyle P}$

## Position vector in different coordinate systems

Cartesian coordinate system

The point described by a position vector can be expressed by the coordinates of a coordinate system, the reference point of the position vector usually being placed in the coordinate origin .

### Cartesian coordinates

Usually the position vector is in Cartesian coordinates in the form

${\ displaystyle {\ vec {r}} = {\ vec {r}} \, (x, y, z) = {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}}}$

Are defined. Therefore the Cartesian coordinates are also the components of the position vector.

### Cylindrical coordinates

The position vector as a function of cylinder coordinates is obtained by converting the cylinder coordinates into the corresponding Cartesian coordinates

${\ displaystyle {\ vec {r}} = {\ vec {r}} \, (\ rho, \ varphi, z) = {\ begin {pmatrix} \ rho \, \ cos \ varphi \\\ rho \, \ sin \ varphi \\ z \ end {pmatrix}}.}$

Here denotes the distance of the point from the -axis, the angle is counted from the -axis in the direction of the -axis. and are therefore the polar coordinates of the point projected orthogonally onto the - plane. ${\ displaystyle \ rho}$${\ displaystyle z}$${\ displaystyle \ phi}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ rho}$${\ displaystyle \ varphi}$${\ displaystyle x}$${\ displaystyle y}$

From a mathematical point of view, the mapping (function) that assigns the Cartesian coordinates of the position vector to the cylinder coordinates is considered here . ${\ displaystyle (\ rho, \ varphi, z)}$${\ displaystyle (x, y, z)}$

### Spherical coordinates

The position vector as a function of spherical coordinates is obtained by converting the spherical coordinates into the corresponding Cartesian coordinates

${\ displaystyle {\ vec {r}} = {\ vec {r}} \, (r, \ theta, \ varphi) = {\ begin {pmatrix} r \, \ sin \ theta \, \ cos \ varphi \ \ r \, \ sin \ theta \, \ sin \ varphi \\ r \, \ cos \ theta \ end {pmatrix}}.}$

Here denotes the distance of the point from the origin (i.e. the length of the position vector), the angle is measured in the - -plane from the -axis in the direction of the -axis, the angle is the angle between the -axis and the position vector. ${\ displaystyle r}$${\ displaystyle \ varphi}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ theta}$${\ displaystyle z}$

## physics

### Celestial mechanics

In order to indicate the position of a celestial body moving on an orbit around a center of gravity , this center of gravity is selected as the origin of the location or radius vector in celestial mechanics . The radius vector then always lies in the direction of the gravitational line . The path of the position vector is called the driving beam . The driving beam plays a central role in Kepler's second law (area theorem) .