# Right-hand rule

In mathematics and physics, the right-hand rule is a mnemonic for understanding the orientation of axes in three-dimensional space. It is a convenient method for determining the direction of the cross product of two vectors. Rather than a mathematical fact, it is a convention. The right-hand rule is closely related to the convention that rotation is represented by a vector oriented such that if one were to view the rotation from the direction towards which the vector points, the rotation appears counter-clockwise.

Left-hand and right-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding one's hands outward and together, palms up, with the thumbs out-stretched to the right and left, and the fingers making a curling motion from straight outward to pointing upward. If the curling motion of the fingers represents a movement from the first (x-axis) to the second (y-axis), then the third (z-axis) can point along either thumb. The rule can be used to find the direction of the magnetic field, rotation, spirals, electromagnetic fields, mirror images, and enantiomers in mathematics and chemistry.

The sequence is often: index finger along the first vector, then middle finger along the second, then thumb along the third. Two other sequences also work because they preserve the cyclic nature of the cross product (and the underlying Levi-Civita symbol):

• Middle finger, thumb, index finger.
• Thumb, index finger, middle finger.

## Coordinates

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 from 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. (If the axes do not have a positive or negative direction, then handedness has no meaning.)

## Rotations

### 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 easy calculations using the vector cross-product. No part of the body is moving in the direction of the axis arrow. By coincidence, 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.

### Helixes and 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

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 (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 and the fingers curl along the orientation of the bounding curve C.

## Electromagnetism

• When electricity flows (with direction given by conventional current) in a long straight wire, it creates a cylindrical magnetic field around the wire according to the right-hand rule. The conventional direction of a magnetic line is given by a compass needle.
• Electromagnet: The magnetic field around a wire is relatively weak. If the wire is coiled into a helix, all the field lines inside the helix point in the same direction and each successive coil reinforces the others. The advance of the helix, the non-circular part of the current, and the field lines all point in the positive z direction. Since there is no magnetic monopole, the field lines exit the +z end, loop around outside the helix, and re-enter at the −z end. The +z end where the lines exit is defined as the north pole. If the fingers of the right hand are curled in the direction of the circular component of the current, the right thumb points to the north pole.
• Lorentz force: If an electric charge moves across a magnetic field, it experiences a force according to Lorentz force, with the direction given by the right-hand rule. If the index finger represents the direction of magnetic field in 3 dimensional space and the middle finger represents the direction of flow of charge (i.e current), the direction of the force on the charge is represented by the thumb. Because the charge is moving, the force causes the particle path to bend. The bending force is computed by the vector cross-product. This means that the bending force increases with the velocity of the particle and the strength of the magnetic field. The force is maximum when the particle direction and magnetic fields are at right angles, is less at any other angle, and is zero when the particle moves parallel to the field.

### Ampère's right-hand grip rule

Ampère's right-hand grip rule (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.

### Application

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.

## Cross products

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:

$\mathbf {F} =q\mathbf {v} \times \mathbf {B}$ (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.

### Applications

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.)

## Meta-mathematical issues

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.