Position (vector)

As a position vector (also the radius vector or position vector ) of a point is called in mathematics and in physics a vector pointing from a fixed reference point to this point (location). In the basic and in synthetic vectors of this geometry may be defined as classes of displacement arrows same or equivalent as a parallel displacement.

Position vectors make it possible to use for the description of points of point sets and pictures of the vector calculus. If you create a Cartesian coordinate system as a basis, one generally chooses the coordinate origin 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 analytic geometry position vectors are used to describe images of an affine or Euclidean space and to describe sets of points (such as lines and planes ) through equations and parametric equations.

In Physics local vectors are used to describe the movement of an (often intended as a point -shaped) body. Local vectors point in coordinate transformations another transformation behavior as covariant vectors.

• 3.1 Cartesian coordinates
• 3.2 cylindrical coordinates
• 3.3 Spherical Coordinates

• 4.1 trajectory
• 4.2 Celestial Mechanics

• 5.1 Relativistic coordinates

Spellings

In geometry, the reference point (origin) is usually denoted by ( for Latin origo ). The notation for the position vector of a point is then:

Occasionally, the small letters with arrow vector are used, which correspond to the capital letters, with which the points are referred to, for example:

In the physics of position vector is referred to as radius vector, and written or not.

The examples and applications in the geometry

Connection vector

For the vector connecting two points with position vectors and the following applies:

Cartesian coordinates

For the coordinates of the position vector of the point with the coordinates of the following applies:

Shift

A shift of the vector is the point from the point. Then for the position vectors:

Rotation around the origin

A rotation in the plane with the center of rotation O by the angle counter-clockwise can be described by means of a rotation matrix in Cartesian coordinates as follows: If the position vector of a point and the position vector of the pixel, then:

Affine

A general affine transformation that maps the point to the point can be represented with the position vectors as follows:

Here, the position vector from, the position vector of a linear map, and a vector that describes a shift. In Cartesian coordinates, the linear map represented by a matrix and we have:

In three-dimensional space, this results in:

Such representations are esauch for other dimensions.

Parametric equation of a straight line

The line through the points and contains exactly the points whose position vector of the representation

Possesses. This is also known by the parametric form of a linear equation.

Normal form of the plane equation

The plane through the point ( base ) with normal vector contains exactly the points whose position vector of the normal equation

Met. Here, the position vector ( support vector machine ) is the Base and the Malpunkt denotes the scalar product.

Position vector in different coordinate systems

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

Cartesian coordinates

Typically, the position vector is in Cartesian coordinates, in the form

Defined. Therefore, the Cartesian coordinates of which are the components of the position vector at the same time.

Cylindrical coordinates

The position vector as a function of cylindrical coordinates is obtained by converting the cylindrical coordinates in the corresponding Cartesian coordinates to

Here, the distance of the point from the axis, the angle is counted from the axis in the direction of the axis. and so are the polar coordinates of the orthogonal to the - plane projected point.

Mathematically, here the picture ( function) is considered, which assigns the cylindrical coordinates, the Cartesian coordinates of the position vector.

Spherical coordinates

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

In this case, (that is, the length of the position vector ) the distance of the point from the origin called the angle is in the - measured level of the axis in the direction of the axis, the angle is the angle between the axis and the radius vector.

Physics

Trajectory

In physics, the location of a point is often mentioned (for example a point mass or center of gravity of a body ) by its position vector. The motion of a point is then described by a function which assigns to each point in time the position vector of the mass point at the time. The curve thus described is also called trajectory or trajectory.

The derivation of this vector valued function with respect to time t yields the velocity vector

If you repeat this derivation, the acceleration vector results

Of valid between times and distance covered for the length:

Celestial mechanics

To change the position of a celestial body that moves in an orbit around a center of gravity, specify is selected in celestial mechanics as the origin of place - or radius vector of this center of gravity. The radius vector is then always in the direction of the line of gravity. The track of the position vector is called the radius vector. The driving beam plays a central role in Kepler's second law ( law of areas ).

Waylet

A path element or line element can be represented as the total differential of the position vector. General results for the vectorial path element when using the coordinates:

With the above equation for the base vectors can be also

. Write The amounts of the derivatives of the position vector to the coordinates are called metric coefficients

This allows you the vectorial path element in the form

Represent. For the previously considered coordinate systems this results in the following display formats:

• Cartesian coordinates:

• Cylindrical coordinates:

• Spherical coordinates:

Relativistic coordinates

In the special relativity theory (SRT ) are space and time as a coherent, four-dimensional pseudoriemannsche manifold, the so-called space-time is described. A point on the manifold, which is defined by three spatial coordinates and a time coordinate is referred to as an event. For each of two events, a line segment DS can be defined by the Minkowski metric, which is proportional to the operating time:

Herein, the Minkowski metric and the four-vector differential.