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In mathematics and physics, the righthand rule is a common mnemonic for understanding orientation of axes in threedimensional space.
Most of the various lefthand and righthand rules arise from the fact that the three axes of threedimensional space have two possible orientations. One can see this by holding one's hands outward and together, palms up, with the fingers curled, and the thumb outstretched. If the curl of the fingers represents a movement from the first or xaxis to the second or yaxis, then the third or zaxis can point along either thumb. Lefthand and righthand rules arise when dealing with coordinate axes. We can use this to find the direction of the magnetic field, rotation, spirals, electromagnetic fields, mirror images, and enantiomers in mathematics and chemistry.
In vector calculus, it is often necessary to relate the normal to a surface to the curve bounding it. For a positivelyoriented curve C bounding a surface S, the normal to the surface n? is defined such that the right thumb points in the direction of n?, and the fingers curl along the orientation of the bounding curve C.
Axis or vector  Two fingers and thumb  Curled fingers 

X, 1, or A  First or index  Fingers extended 
Y, 2, or B  Second finger or palm  Fingers curled 90° 
Z, 3, or C  Thumb  Thumb 
Coordinates are usually righthanded.
For righthanded coordinates the right thumb points along the Z axis in the positive direction and the curl of the fingers represents a motion from the first or X axis to the second or Y axis. When viewed from the top or Z axis the system is counterclockwise.
For lefthanded coordinates the left thumb points along the Z axis in the positive direction and the curled fingers of the left hand represent a motion from the first or X axis to the second or Y axis. When viewed from the top or Z axis the system is clockwise.
Interchanging the labels of any two axes reverses the handedness. Reversing the direction of one axis (or of all three axes) also reverses the handedness. (If the axes do not have a positive or negative direction then handedness has no meaning.) Reversing two axes amounts to a 180° rotation around the remaining axis.^{[1]}
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 righthand 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 in a prograde direction according to the righthand rule. This causes the Sun, Moon, and stars to appear to revolve westward according to the lefthand rule.
A helix is a curved line formed by a point rotating around a center while the center moves up or down the Zaxis. Helices are either right or lefthanded, curled fingers giving the direction of rotation and thumb giving the direction of advance along the Zaxis.
The threads of a screw are a helix and therefore screws can be right or lefthanded. The rule is this: if a screw is righthanded (most screws are) point your right thumb in the direction you want the screw to go and turn the screw in the direction of your curled right fingers.
Ampère's righthand grip rule^{[2]} (also called righthand screw rule, coffeemug rule or the corkscrewrule) 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.
AndréMarie Ampère, a French physicist and mathematician, for whom the rule was named, was inspired by Hans Christian Ørsted, another physicist who experimented with magnet needles. Ørsted observed that the needles swirled when in the proximity of an electric currentcarrying wire, and concluded that electricity could create magnetic fields.
This rule is used in two different applications of Ampère's circuital law:
The cross product of two vectors is often taken in physics and engineering. For example, in statics and dynamics, torque is the cross product of lever length and force, while angular momentum is the cross product of linear momentum and distance. In electricity and magnetism, the force exerted on a moving charged particle when moving in a magnetic field B is given by:
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.^{[1]}
The right hand rule is in widespread use in physics. A list of physical quantities whose directions are related by the righthand rule is given below. (Some of these are related only indirectly to cross products, and use the second form.)