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Concept
Right-hand rule
Summary
In mathematics and physics, the right-hand rule is a common mnemonic for understanding the orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of the cross product of two vectors. Rather than a mathematical fact, it is a convention, closely related to the convention that rotation around a vertical axis is positive if it is counterclockwise and negative if it is clockwise.
Most left-hand and right-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. One can see this 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. Left-hand and right-hand rules arise when dealing with coordinate axes. 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 (e.g., see the ninth series of the Swiss 200-francs banknote).
In vector calculus, it is necessary to relate the normal vector to a surface to the curve bounding it. For a positively-oriented curve C, bounding a surface S, the normal to the surface n̂ is defined such that the right thumb points in the direction of n̂, and the fingers curl along the orientation of the bounding curve C.
For right-handed coordinates, the right thumb points along the z-axis in the positive direction and the curling motion of the fingers of the right hand represents a motion from the first or x-axis to the second or y-axis.
Official source
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