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

In mathematics and physics, the right-hand 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.

Contents

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.

 \vec{a} \times \vec{b} = \vec{c}
  • With the thumb, index, and middle fingers at right angles to each other (with the index finger pointed straight), the middle finger points in the direction of c when the thumb represents a and the index finger represents b.

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]

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.[4] 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

See also

Notes

  1. ^ Fleming, John Ambrose (1902). Magnets and Electric Currents, 2nd Edition. London: E.& F. N. Spon. pp. 173–174. http://books.google.com/books?id=ASUYAAAAYAAJ&pg=PA173. 
  2. ^ "Right and left hand rules". Tutorials, Magnet Lab U.. National High Magnetic Field Laboratory. http://www.magnet.fsu.edu/education/tutorials/java/handrules/index.html. Retrieved 2008-04-30. 
  3. ^ PHYS345 Introduction to the Right Hand Rule, George Watson, University of Delaware, 1998
  4. ^ Wilson, Adam (2008). "Hand Rules". Course outline, EE2683 Electric Circuits and Machines. Faculty of Engineering, Univ. of New Brunswick. http://www.ece.unb.ca/Courses/EE2683/AW/hand_rules.pdf. Retrieved 2008-08-11. 

External links

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Simple English

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 [[Right-hand fist rule] that is used for magnetic fields and things that rotate.

  1. Start by closing your right hand and stick out your pointer finger.
  2. Stick your thumb straight up like a your making the sign for a gun.
  3. If you point your "gun" straight ahead, stick out your middle finger so that it points left and all your fingers are at right angles to each other.

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 \vec{thumb} \times \vec{pointer} = \vec{middle}.


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