page_title: Cartesian coordinate system
section_title: In two dimensions
hierarchical_section_title: Cartesian coordinate system / Orientation and handedness / In two dimensions
caption_attribution_description: Deutsch: Rechte Hand, die die Rechte-Hand-Regel erklärt. a,b,c / x,y,z sind Vektoren, l,B,F stehen für die Formel F &#x2192; = i ( l &#x2192; &#x00D7; B &#x2192; ) {\displaystyle {\vec {F}}=i({\vec {l}}\times {\vec {B}})} . F ist hierbei die Lorentzkraft, die auf einen stromdurchflossenen Leiter l ausgeübt wird, wenn auf ihn ein B-Feld einwirkt. Der Daumen zeigt in Richtung der Ursache (technische Stromrichtung I bzw. Bewegungsrichtung der positiven Ladung q also von + nach -). Der Zeigefinger zeigt in Richtung der magnetischen Feldlinien, also der Vermittlung (B-Feld ) von Nord nach Süd. Der Mittelfinger entspricht der Wirkung (Kraft). Bei negativen bewegten Ladungen gilt dies analog, allerdings für die linke Hand.
context_page_description: A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
  One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
  The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
context_section_description: Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane.
  The usual way of orienting the plane, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the "first" and the y-axis the "second" axis), is considered the positive or standard orientation, also called the right-handed orientation.
  A commonly used mnemonic for defining the positive orientation is the right-hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.
  The other way of orienting the plane is following the left hand rule, placing the left hand on the plane with the thumb pointing up.
  When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis.
  Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation, but switching both will leave the orientation unchanged.
caption: The right-hand rule
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