Vector size

A vector quantity or directed quantity is a physical quantity which - in contrast to the scalar quantities - has a directional character. Typical vector quantities are the kinematic quantities speed and acceleration , the dynamic quantities momentum and force or angular momentum and torque as well as the field strengths of the electric and magnetic fields of electrodynamics . Vector quantities are treated like geometric vectors , both graphically and mathematically , although some special features must be observed.

presentation

The force as a vector quantity is illustrated by an arrow. The vector can be moved along its line of action.

Vector quantities are usually indicated by an arrow above the symbol ( ) or by bold type ( ). The corresponding size symbol without identification stands for the amount of the size: or . In drawings, the vector size is represented by an arrow, the length of which stands for the amount of the size. ${\displaystyle {\vec {a}}}$ ${\displaystyle \mathbf {a} }$ ${\displaystyle a:=|{\vec {a}}|}$ ${\displaystyle a:=|\mathbf {a} |}$

Mathematical properties

In contrast to geometric vectors, a vector quantity does not represent a spatial displacement. So are z. For example, forces are not elements of local space, but rather their own vector space , even if they are entered in spatial sketches for illustration. In other words: although they are shown as directed routes, they usually do not have the dimension of a path.

The vector space in which a vector quantity lives is generally assigned to a point in space. This can be expressed using an index . A vector quantity at another point is not an element of vector space , but lives in its own vector space . Two vector quantities can only be added if they are elements of the same vector space. This is done through a parallel shift . For example, a force vector can be shifted along its line of action (see figure). ${\displaystyle V}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle P}$ ${\displaystyle {\vec {a}}_{P}\in V_{P}}$ ${\displaystyle Q}$ ${\displaystyle V_{P}}$ ${\displaystyle V_{Q}}$

If there is a function that assigns a vector quantity to every point in space, the function is called a vector field .

Many physical problems can be described in three-dimensional Euclidean space . Such a vector quantity can therefore be described by a vector from a vector space with dimension  3. In the theory of relativity , to specify a direction in space-time , a direction in time is also specified, which is why four-vectors from a four-dimensional vector space are usually used here .

Depending on the behavior under spatial reflection, a distinction is made between axial and polar vectors . A vector quantity given by an axial vector maintains its direction, while a vector quantity given by a polar vector reverses its direction. The behavior under an ordinary coordinate transformation described below is the same for both.

Coordinates and components

A vector quantity can be described by its coordinates , i. H. by a tuple of numbers, which characterizes the orientation of the size in space. Frequently Cartesian coordinates used. In order to reproduce the directional character of a vector quantity, the representation as a column vector is suitable :

${\displaystyle {\vec {a}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}}$

Instead of 1, 2 and 3, the coordinate axes are often referred to as , and . The axis is usually the vertical axis while the plane extends horizontally. Sometimes other coordinate systems such as spherical coordinates make more sense. In this case, the amount of the size is indicated on the one hand and on the other hand, the direction through the two angles and . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle xy}$ ${\displaystyle \vartheta }$ ${\displaystyle \varphi }$

The component notation is also used. The with are the coordinates of the vector with respect to a previously established basis . The size can then be written as the sum of its components : ${\displaystyle a_{i}}$ ${\displaystyle i\in \{1,2,3\}}$ ${\displaystyle \{{\vec {e}}_{1},{\vec {e}}_{2},{\vec {e}}_{3}\}}$ ${\displaystyle a_{i}{\vec {e}}_{i}}$

${\displaystyle {\vec {a}}=\sum _{i=1}^{3}a_{i}{\vec {e}}_{i}}$

For an orthonormal basis , conversely, the coordinates are obtained from the scalar product

${\displaystyle a_{i}={\vec {a}}\cdot {\vec {e}}_{i}}$.

The coordinates of a vector quantity are different depending on the choice of the basis vectors. The coordinates in the unprimed coordinate system are linked to the coordinates in the primed coordinate system via the relation ${\displaystyle a_{i}}$ ${\displaystyle a'_{i}}$

${\displaystyle {\vec {a}}=\sum _{i=1}^{3}a_{i}{\vec {e}}_{i}=\sum _{i=1}^{3}a'_{i}{\vec {e}}'_{i}}$

together. Usually a coordinate base is used in which the base vectors point "in the direction of the coordinates" . Using superscript and subscript indices, covariant and contravariant vectors can be distinguished. The transformation behavior of a vector quantity under a coordinate transformation corresponds to a tensor of level 1, a scalar corresponds to a tensor of level 0. Vector quantities can therefore generally be defined via their transformation behavior under coordinate transformations. The transformation of the basis vectors is through ${\displaystyle \textstyle {\vec {e}}_{i}={\frac {\partial }{\partial x^{i}}}}$ ${\displaystyle x^{i}}$

${\displaystyle {\vec {e}}'_{i}=\sum _{j=1}^{3}{\frac {\partial x^{j}}{\partial x'^{i}}}{\vec {e}}_{j}}$

Are defined. For example, the base vector is at a transformation of the coordinates to by the partial derivatives of the coordinate functions , and optionally: ${\displaystyle {\vec {e}}_{r}}$ ${\displaystyle (x,y,z)}$ ${\displaystyle (r,\vartheta ,\varphi )}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

${\displaystyle {\vec {e}}_{r}={\frac {\partial x(r,\vartheta ,\varphi )}{\partial r}}{\vec {e}}_{x}+{\frac {\partial y(r,\vartheta ,\varphi )}{\partial r}}{\vec {e}}_{y}+{\frac {\partial z(r,\vartheta ,\varphi )}{\partial r}}{\vec {e}}_{z}}$

The weight force is to serve as an example for the three types of representation , which has the amount , where is the mass of the body and the acceleration due to gravity . ${\displaystyle F=mg}$ ${\displaystyle m}$ ${\displaystyle g}$

• In Cartesian coordinates: (in a homogeneous gravitational field). ${\displaystyle {\vec {F}}={\begin{pmatrix}0\\0\\-mg\end{pmatrix}}}$
• In spherical coordinates: (in the real gravitational field of the earth, with the center of the earth being the origin of the coordinates). ${\displaystyle {\vec {F}}=m\,{\vec {g}}(r,\vartheta ,\varphi )}$
• In component notation: (in the homogeneous gravitational field). Here the Kronecker delta was used, which in this case disappears for everyone . ${\displaystyle {\vec {F}}=\sum _{i=1}^{3}F_{i}=\sum _{i=1}^{3}-\delta _{i3}mg{\vec {e}}_{i}=-mg{\vec {e}}_{3}}$ ${\displaystyle i\neq 3}$

Constraints can reduce the number of coordinates required. If a physical problem is limited to one plane, a two-dimensional coordinate system is sufficient. In the one-dimensional case, the directional character of the vector quantity can only be recognized by its sign.

Web links

• Vector and Tensor Algebra (including Column and Matrix Notation). - Overview of calculation rules and notations for vectors (PDF, 266 kB, English).

Individual evidence

1. ↑ Rana & Joag: Classical Mechanics . Tata McGraw-Hill Education, 2001, ISBN 0-07-460315-9 , pp. 559 ( limited preview in Google Book search).