# Right-hand rule

**Editor-In-Chief:** C. Michael Gibson, M.S., M.D. [1]

In mathematics and physics, the **right-hand rule** is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented by British physicist Ambrose Fleming in the early 1900s^{[1]}

When choosing three vectors that must be at right angles to each other, there are two distinct solutions, so when expressing this idea in mathematics, one must remove the ambiguity of which solution is meant.

There are variations on the mnemonic depending on context, but all variations are related to the one idea of choosing a convention.

### Direction associated with an ordered pair of directions

One form of the right-hand rule is used in situations in which an ordered operation must be performed on two vectors *a* and *b* that has a result which is a vector *c* perpendicular to both *a* and *b*. The most common example is the vector cross product. The right-hand rule imposes the following procedure for choosing one of the two directions.

- <math> \vec{a} \times \vec{b} = \vec{c}</math>

- Hold the thumb, index and middle fingers at right angles to each other. Make sure that the thumb and index finger form an "L" or a gun shape. The middle finger is the direction of
*c*when the thumb represents*a*and the index finger represents*b*.

### Direction associated with a rotation

A different form of the right-hand rule is used in situations where a vector must be assigned to the *rotation* of a body, a magnetic field or a fluid. Alternatively, when a rotation is specified by a vector, and it is necessary to understand the way in which the rotation occurs, the right-hand rule is applicable.

In this form, the fingers of the right hand are curled to match the curvature and direction of the motion or the magnetic field. The thumb indicates the direction of the vector.

### Applications

The first form of the rule is used to determine the direction of the cross product of two vectors. This leads to widespread use in physics, wherever the cross product occurs. 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.)

- The angular velocity of a rotating object and the rotational velocity of any point on the object
- A torque, the force that causes it, and the position of the point of application of the force
- A magnetic field, the position of the point where it is determined, and the electric current (or change in electric flux) that causes it
- A magnetic field in a coil of wire and the electric current in the wire
- The force of a magnetic field on an object, the magnetic field itself, and the velocity of the object
- The vorticity at any point in the field of flow of a fluid
- The induced current from motion in a magnetic field (known as Fleming's right hand rule)

Fleming's left hand rule is a rule for finding the direction of the thrust on a conductor carrying a current in a magnetic field.

## Left handedness

In certain situations, it may be useful to use the opposite convention, where one of the vectors is reversed and so creates a left-handed triad instead of a right-handed triad.

An example of this situation is for left-handed materials. Normally, for an electromagnetic wave, the electric and magnetic fields, and the direction of propagation of the wave obey the right-hand rule. However, left-handed materials have special properties - the negative refractive index. It makes the direction of propagation point in the opposite direction.

De Graaf's translation of Fleming's left-hand rule - which uses thrust, field and current - and the right-hand rule, is the FBI rule. The FBI rule changes Thrust into F (Lorentz force), B (direction of the magnetic field) and I (current). The FBI rule is easily remembered by US citizens because of the commonly known abbreviation for the Federal Bureau of Investigation.

## Symmetry

Vector | Right-Hand | Right-Hand | Right-Hand | Left-Hand | Left-Hand | Left-Hand |
---|---|---|---|---|---|---|

a, x or I | Thumb | Fingers or Palm | First or Index | Thumb | Fingers or Palm | First or Index |

b, y or B | First or Index | Thumb | Fingers or Palm | Fingers or Palm | First or Index | Thumb |

c, z or F | Fingers or Palm | First or Index | Thumb | First or Index | Thumb | Fingers or Palm |

## See also

- Chirality (mathematics)
- Cross product
- Right hand grip rule
- Curl
- Pseudovector
- Improper rotation
- Reflection (mathematics)
- Fleming's left hand rule
- Vorticity

## External links

- A demonstration of the right-hand rule at physics.syr.edu
- Definition at mathworld.wolfram.com
- Comic depicting other alternatives to the right-hand rule

cs:Pravidlo pravé ruky de:Drei-Finger-Regel it:Regola della mano destra he:כלל יד ימין nl:Rechterhandregel sv:Högerhandsregeln

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- ↑ "Right and left hand rules".
*Tutorials, Magnet Lab U*. National High Magnetic Field Laboratory. Retrieved 2008-04-30.