Right-hand rule

The left-handed orientation is shown on the left, and the right-handed on the right.
Use of right hand.

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.

${\displaystyle {\vec {a}}\times {\vec {b}}={\vec {c}}}$
• 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

Vector assigned to 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.)

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.

Fleming's left hand rule

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