Right-hand rule (original) (raw)

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Mnemonic for understanding orientation of vectors in 3D space

This article is about three-dimensional vector geometry. For the maze-solving technique, see Wall follower. For the traffic rule, see Priority to the right.

In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic 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 of quaternions, 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 of two vector quaternions yields a quaternion comprising both scalar and vector components.[1] Josiah Willard Gibbs recognized that treating these components separately, as dot and cross 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.

Right-hand rule for cross product

The cross product of vectors a → {\displaystyle {\vec {a}}} {\displaystyle {\vec {a}}} and b → {\displaystyle {\vec {b}}} {\displaystyle {\vec {b}}} is a vector perpendicular to the plane spanned by a → {\displaystyle {\vec {a}}} {\displaystyle {\vec {a}}} and b → {\displaystyle {\vec {b}}} {\displaystyle {\vec {b}}} with the direction given by the right-hand rule: If you put the index of your right hand on a → {\displaystyle {\vec {a}}} {\displaystyle {\vec {a}}} and the middle finger on b → {\displaystyle {\vec {b}}} {\displaystyle {\vec {b}}}, then the thumb points in the direction of a → × b → {\displaystyle {\vec {a}}\times {\vec {b}}} {\displaystyle {\vec {a}}\times {\vec {b}}}.[3]

Fleming's right hand rule

The right-hand rule in physics was introduced in the late 19th century by John Fleming in his book Magnets and Electric Currents.[4] 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 of magnetic flux, the direction of which is represented by the direction of the forefinger, 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."[4]

Left-handed coordinates on the left,
right-handed coordinates on the right.

For right-handed coordinates, use the right hand.For left-handed coordinates, use the left hand.

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

For right-handed coordinates, if the thumb of a person's right hand points along the _z_-axis in the positive direction (third coordinate vector), then the fingers curl from the positive _x_-axis (first coordinate vector) toward the positive _y_-axis (second coordinate vector). When viewed at a position along the positive _z_-axis, the ¼ turn from the positive x- to the positive _y-_axis is counter-clockwise.

For left-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn is clockwise.

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 operations can be composed to give repeated changes of handedness.[5] (If the axes do not have a positive or negative direction, then handedness has no meaning.)

Conventional direction of the axis of a rotating body

In mathematics, a rotating body is commonly represented by a pseudovector along the axis of rotation. The length of the vector gives the speed 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 to appear to revolve westward according to the left-hand rule.

Left- and right-handed screws

A helix is a curved line formed by a point rotating around a center while the center moves up or down the _z_-axis. Helices are either right or left handed with curled fingers giving the direction of rotation and thumb giving the direction of advance along the _z_-axis.

The threads of a screw 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.

Curve orientation and normal vectors

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In vector calculus, it is necessary to relate a normal vector of a surface to the boundary curve of the surface. Given a surface S with a specified normal direction n̂ (a choice of "upward direction" with respect to S), the boundary curve C around S is defined to be positively oriented provided that the right thumb points in the direction of n̂ and the fingers curl along the orientation of the bounding curve C.

Ampère's right-hand grip rule

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Prediction of direction of field (B), given that the current I flows in the direction of the thumb

Finding direction of magnetic field (B) for an electrical coil

Ampère's right-hand grip rule,[6] also called the right-hand screw rule, coffee-mug rule or the corkscrew-rule; is used either when a vector (such as the Euler vector) must be defined to represent the rotation of a body, a magnetic field, or a fluid, or vice versa, when it is necessary to define a rotation vector to understand how rotation occurs. It reveals a connection between the current and the magnetic field lines in the magnetic field that the current created. Ampère was inspired by fellow physicist Hans Christian Ørsted, who observed that needles swirled when in the proximity of an electric current-carrying wire and concluded that electricity could create magnetic fields.

This rule is used in two different applications of Ampère's circuital law:

  1. An electric current passes through a straight wire. When the thumb is pointed in the direction of conventional current (from positive to negative), the curled fingers will then point in the direction of the magnetic flux lines around the conductor. The direction of the magnetic field (counterclockwise rotation instead of clockwise rotation of coordinates when viewing the tip of the thumb) is a result of this convention and not an underlying physical phenomenon.
  2. An electric current passes through a solenoid, resulting in a magnetic field. When wrapping the right hand around the solenoid with the fingers in the direction of the conventional current, the thumb points in the direction of the magnetic north pole.

Illustration of the right-hand rule on the ninth series of the Swiss 200-francs banknote.

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:

F = q v × B {\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} } {\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} } (vector cross product)

The direction of the cross product may be found by application of the right-hand rule as follows:

  1. The index finger points in the direction of the velocity vector v.
  2. The middle finger points in the direction of the magnetic field vector B.
  3. The thumb points in the direction of the cross product F.

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.[5]

The right-hand rule has widespread use in physics. 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 any mathematical axioms. Rather, the definition depends on chiral 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 the weak force.

  1. ^ Hamilton, William Rowan (1853). Lectures on quaternions. unknown library. Dublin.
  2. ^ Chappell, James M.; Iqbal, Azhar; Hartnett, John G.; Abbott, Derek (2016). "The Vector Algebra War: A Historical Perspective". IEEE Access. 4: 1997–2004. arXiv:1509.00501. Bibcode:2016IEEEA...4.1997C. doi:10.1109/access.2016.2538262. ISSN 2169-3536.
  3. ^ Hubbard, John H. (John Hamal) (2009). Vector calculus, linear algebra, and differential forms : a unified approach. Internet Archive. Ithaca, NY : Matrix Editions. ISBN 978-0-9715766-5-0.
  4. ^ a b Fleming, J. A. (John Ambrose) (1902). Magnets and electric currents. An elementary treatise for the use of electrical artisans and science teachers. Harvard University. London, E. & F.N. Spon, limited; New York, Spon & Chamberlain.
  5. ^ a b Watson, George (1998). "PHYS345 Introduction to the Right Hand Rule". udel.edu. University of Delaware.
  6. ^ IIT Foundation Series: Physics – Class 8, Pearson, 2009, p. 312.