Legal system (mathematics)

Axis orientation and sense of rotation of left-handed and right-handed coordinate systems

As a legal system or right-handed coordinate system in which are mathematics and physics certain systems (with a fixed order) of two vectors in the plane or three vectors in the space designated.

Legal system in the plane

A right system in the plane are two vectors , in which from the shortest path by rotating counterclockwise , i.e. H. in the mathematically positive sense of rotation . ${\displaystyle {\vec {x}},{\vec {y}}}$ ${\displaystyle {\vec {y}}}$ ${\displaystyle {\vec {x}}}$

Legal system in space

A legal system in three-dimensional space are three vectors and , if viewed from the end point of the vector , the vectors form a legal system in the plane. ${\displaystyle {\vec {x}},{\vec {y}}}$ ${\displaystyle {\vec {z}}}$ ${\displaystyle {\vec {z}}}$ ${\displaystyle {\vec {x}},{\vec {y}}}$

Legal system in ${\displaystyle \mathbb {R} ^{n}}$

A legal system is generally an ordered tuple of column vectors of the dimension such that the determinant of the matrix with the column vectors is positive. ${\displaystyle ({\vec {x}}_{1},\dotsc ,{\vec {x}}_{n})}$ ${\displaystyle n}$ ${\displaystyle {\vec {x}}_{1},\dotsc ,{\vec {x}}_{n}}$

For and this is equivalent to the above definitions. ${\displaystyle n=2}$ ${\displaystyle n=3}$

Link systems

For links systems or left-handed coordinate systems , respectively the reverse is true. In the plane the first vector goes by rotating clockwise , i. H. mathematically negative direction of rotation emerges from the second vector on the shortest path, just as it is itself converted into the second vector on the shortest path by clockwise rotation .

A left system in a vector space is an ordered tuple of column vectors in which the associated matrix has a negative determinant. Accordingly, a left system in three-dimensional space is an ordered triple of vectors for which the above late product is negative.

regulate

The following rules can be used to determine whether three vectors form a right or left system:

• With the three-finger rule of the right hand (also called the right-hand rule ): If the splayed thumb points in the direction of the first vector and the extended index finger towards the second vector, the middle finger splayed out at right angles to the thumb and forefinger points in a right-hand system in the direction of the third vector (this also works if the fingers or vectors are cyclically swapped :) . ${\displaystyle x-y-z,y-z-x,z-x-y}$
• With the screw or corkscrew rule : If the first vector is rotated so that it is transferred to the second vector in the shortest possible way, a screw with a right-hand thread rotated in the same sense moves in the direction of the third , provided that all three vectors form a right system Vector.

For 2-dimensional systems, a rule analogous to the three-finger rule can be formulated as follows: If the thumb of the right hand opened upwards points in the positive direction, in a right-handed system all other fingers point in the positive direction - do if they don't, the system is left-handed. ${\displaystyle x}$ ${\displaystyle y}$

Examples

• The axes of the three-dimensional Cartesian coordinate system form a right-hand system in its usual axis orientation (e.g. -axis to the viewer, -axis to the right, -axis up; also -axis to the right, -axis in perspective and -axis up) . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$
• The geodetic coordinate system , on the other hand, follows the direction of rotation of the compass , a left system.
• Another widespread link system is that of the pixel coordinates in graphics programs, in which the coordinate origin (0 | 0) is usually in the upper left corner of the screen and the coordinates (graphic columns) count from there to the right, while the coordinates (graphic lines) count downwards The coordinates of an image point must first always be subjected to a corresponding coordinate transformation in order to be displayed on the screen . ${\displaystyle x}$ ${\displaystyle y}$
• When a body rotates, the radial vector, the tangential velocity and the angular momentum form a right system.
• When a current-carrying conductor is deflected in a magnetic field ( ladder swing experiment), the technical direction of the current , the magnetic field lines and the direction of action of the Lorentz force form a right system.

References and comments

1. ^ Walter Gellert, Herbert Küstner, Manfred Hellwich, Herbert Kästner (eds.): Small encyclopedia of mathematics ; Leipzig 1970, pp. 342-343.
2. ↑ Almost all screws used in practice have a right - hand thread - those with a left- hand thread, on the other hand, are rarely used, e.g. B. in turnbuckles