In mathematics and physics, the righthand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 1800s.^{[1]}^{[2]}
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.
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One form of the righthand 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 righthand rule imposes the following procedure for choosing one of the two directions.
Other (equivalent) finger assignments are possible. For example, the first (index) finger can represent a, the first vector in the product; the second (middle) finger, b, the second vector; and the thumb, c, the product.^{[3]}
A different form of the righthand rule is used in situations where a vector must be assigned to the rotation of a body, a magnetic field or a fluid.^{[4]} Alternatively, when a rotation is specified by a vector, and it is necessary to understand the way in which the rotation occurs, the righthand 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.
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 righthand 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.
In certain situations, it may be useful to use the opposite convention, where one of the vectors is reversed and so creates a lefthanded triad instead of a righthanded triad.
An example of this situation is for lefthanded materials. Normally, for an electromagnetic wave, the electric and magnetic fields, and the direction of propagation of the wave obey the righthand rule. However, lefthanded 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 lefthand rule  which uses thrust, field and current  and the righthand 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.
Vector  RightHand  RightHand  RightHand  LeftHand  LeftHand  LeftHand 

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 
The Right Hand Rule is a convention in vector math. It helps you remember direction when vectors get cross multiplied. There is another rule called the [[Righthand fist rule] that is used for magnetic fields and things that rotate.
If you have two vectors that you want to cross multiply, you can figure out the direction of the vector that comes out by pointing your thumb in the direction of the first vector and your pointer in the direction of the second vector. Your middle finger will point the direction of the cross product.
Remember that when you change the order that vectors get cross multiplied, the result goes in the opposite direction. So it's important to make sure that you go in the order of $\backslash vec\{thumb\}\; \backslash times\; \backslash vec\{pointer\}\; =\; \backslash vec\{middle\}$.
