Inmathematics andphysics, theright-hand rule is aconvention and amnemonic, utilized to define theorientation ofaxes inthree-dimensional space and to determine the direction of thecross product of twovectors, as well as to establish the direction of the force on acurrent-carrying conductor in amagnetic field.
The various right- and left-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding your hands together with palms up and fingers curled. If the curl of the fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either right thumb or left thumb.
The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dimensions.William Rowan Hamilton, recognized for his development ofquaternions, a mathematical system for representing three-dimensional rotations, is often attributed with the introduction of this convention. In the context of quaternions, the Hamiltonian product oftwo vector quaternions yields a quaternion comprising bothscalar andvector components.[1]Josiah Willard Gibbs recognized that treating these components separately, asdot andcross product, simplifies vector formalism. Following a substantial debate,[2] the mainstream shifted from Hamilton's quaternionic system to Gibbs' three-vectors system. This transition led to the prevalent adoption of the right-hand rule in the contemporary contexts. In specific, Gibbs outlines his intention for establishing a right-handed coordinate system in his pamphlet on vector analysis.[3] In Article 11 of the pamphlet, Gibbs states "The letters,, and are appropriated to the designation of anormal system of unit vectors, i.e., three unit vectors, each of which is at right angles to the other two ... We shall always suppose that is on the side of the plane on which a rotation from to (through one right angle) appears counter-clockwise." While Gibbs did not use the termright-handed in his discussion, his instructions for defining the normal coordinate orientation are a clear statement of his intent for coordinates that follow the right-hand rule.
The cross product of vectors and is a vector perpendicular to the plane spanned by and with the direction given bythe right-hand rule: If you put theindex of your right hand on and themiddle finger on, then thethumb points in the direction of.[4]
The right-hand rule in physics was introduced in the late 19th century byJohn Fleming in his book Magnets and Electric Currents.[5] Fleming described the orientation of the induced electromotive force by referencing the motion of the conductor and the direction of the magnetic field in the following depiction: “If a conductor, represented by the middle finger, be moved in a field ofmagnetic flux, the direction of which is represented by the direction of theforefinger, the direction of this motion, being in the direction of the thumb, then the electromotive force set up in it will be indicated by the direction in which the middle finger points."[5]
| Axis/vector | Two fingers and thumb | Curled fingers |
|---|---|---|
| x (or first vector) | First or index | Fingers extended |
| y (or second vector) | Second finger or palm | Fingers curled 90° |
| z (or third vector) | Thumb | Thumb |
Forright-handed coordinates, if the thumb of a person's right hand points along thez-axis in the positive direction (third coordinate vector), then the fingers curl from the positivex-axis (first coordinate vector) toward the positivey-axis (second coordinate vector). When viewed at a position along the positivez-axis, the ¼ turn from the positivex- to the positivey-axis iscounter-clockwise.
Forleft-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn isclockwise.
Interchanging the labels of any two axes reverses the handedness. Reversing the direction of one axis (or three axes) also reverses the handedness. Reversing two axes amounts to a 180° rotation around the remaining axis, also preserving the handedness. These operationscan be composed to give repeated changes of handedness.[6] (If the axes do not have a positive or negative direction, then handedness has no meaning.)
In mathematics, a rotating body is commonly represented by apseudovector along the axis ofrotation. The length of the vector gives thespeed of rotation and the direction of the axis gives the direction of rotation according to the right-hand rule: right fingers curled in the direction of rotation and the right thumb pointing in the positive direction of the axis. This allows some simple calculations using the vector cross-product. No part of the body is moving in the direction of the axis arrow. If the thumb is pointing north,Earth rotates according to the right-hand rule (prograde motion). This causes the Sun, Moon, and stars toappear to revolve westward according to the left-hand rule.
Ahelix is a curved line formed by a point rotating around a center while the center moves up or down thez-axis. Helices are either right or left handed with curled fingers giving the direction of rotation and thumb giving the direction of advance along thez-axis.
The threads of ascrew are helical and therefore screws can be right- or left-handed. To properly fasten or unfasten a screw, one applies the above rules: if a screw is right-handed, pointing one's right thumb in the direction of the hole and turning in the direction of the right hand's curled fingers (i.e. clockwise) will fasten the screw, while pointing away from the hole and turning in the new direction (i.e. counterclockwise) will unfasten the screw.
Invector calculus, it is necessary to relate anormal vector of a surface to the boundary curve of the surface. Given a surfaceS with a specified normal directionn̂ (a choice of "upward direction" with respect toS), the boundary curveC aroundS is defined to bepositively oriented provided that the right thumb points in the direction ofn̂ and the fingers curl along the orientation of the bounding curveC.
Ampère's right-hand grip rule,[7] also called theright-hand screw rule,coffee-mug rule or thecorkscrew-rule; is used either when avector (such as theEuler vector) must be defined to represent therotation of a body, a magnetic field, or a fluid, or vice versa, when it is necessary to define arotation vector to understand how rotation occurs. It reveals a connection between the current and themagnetic field lines in the magnetic field that the current created. Ampère was inspired by fellow physicistHans Christian Ørsted, who observed that needles swirled when in the proximity of anelectric current-carrying wire and concluded that electricity could createmagnetic fields.
This rule is used in two different applications ofAmpère's circuital law:
The cross product of two vectors is often taken in physics and engineering. For example, as discussed above, the force exerted on a moving charged particle when moving in a magnetic field B is given by the magnetic term of Lorentz force:
The direction of the cross product may be found by application of the right-hand rule as follows:
For example, for a positively charged particle moving to the north, in a region where the magnetic field points west, the resultant force points up.[6]
The right-hand rule has widespread use inphysics. A list of physical quantities whose directions are related by the right-hand rule is given below. (Some of these are related only indirectly to cross products, and use the second form.)
Unlike most mathematical concepts, the meaning of a right-handed coordinate system cannot be expressed in terms of anymathematical axioms. Rather, the definition depends onchiral phenomena in the physical world, for example the culturally transmitted meaning of right and left hands, a majority human population with dominant right hand, or certain phenomena involving theweak force.
