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Right-hand rule
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== History == The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dimensions. [[William Rowan Hamilton]], recognized for his development of [[quaternion]]s, a mathematical system for representing three-dimensional rotations, is often attributed with the introduction of this convention. In the context of quaternions, the Hamiltonian product of [[Quaternions and spatial rotation|two vector quaternions]] yields a quaternion comprising both [[Scalar (mathematics)|scalar]] and [[Vector space|vector]] components.<ref>{{Cite book |last=Hamilton |first=William Rowan |url=http://archive.org/details/bub_gb_TCwPAAAAIAAJ |title=Lectures on quaternions |date=1853 |publisher=Dublin |others=unknown library}}</ref> [[Josiah Willard Gibbs]] recognized that treating these components separately, as [[Dot product|dot]] and [[Cross product|cross]] product, simplifies vector formalism. Following a substantial debate,<ref>{{Cite journal |last1=Chappell |first1=James M. |last2=Iqbal |first2=Azhar |last3=Hartnett |first3=John G. |last4=Abbott |first4=Derek |date=2016 |title=The Vector Algebra War: A Historical Perspective |url=http://dx.doi.org/10.1109/access.2016.2538262 |journal=IEEE Access |volume=4 |pages=1997–2004 |doi=10.1109/access.2016.2538262 |issn=2169-3536|arxiv=1509.00501 |bibcode=2016IEEEA...4.1997C }}</ref> the mainstream shifted from Hamilton's quaternionic system to Gibbs' three-vectors system. This transition led to the prevalent adoption of the right-hand rule in the contemporary contexts. In specific, Gibbs outlines his intention for establishing a right-handed coordinate system in his pamphlet on vector analysis.<ref>{{Cite book |last=Gibbs |first=Josiah Willard |url=https://archive.org/details/elementsvectora00gibb |title=Elements of Vector Analysis: Arranged for the Use of Students in Physics |date=1881 |publisher=Tuttle, Morehouse & Taylor | place=New Haven, CT}}</ref> In Article 11 of the pamphlet, Gibbs states "The letters <math>i</math>, <math>j</math>, and <math>k</math> are appropriated to the designation of a ''normal system of unit vectors'', i.e., three unit vectors, each of which is at right angles to the other two ... We shall always suppose that <math>k</math> is on the side of the <math>i-j</math> plane on which a rotation from <math>i</math> to <math>j</math> (through one right angle) appears counter-clockwise." While Gibbs did not use the term ''right-handed'' in his discussion, his instructions for defining the normal coordinate orientation are a clear statement of his intent for coordinates that follow the right-hand rule. [[File:Right-hand rule for cross product.png|thumb|Right-hand rule for cross product]] The cross product of vectors <math>\vec{a}</math> and <math>\vec{b}</math> is a vector perpendicular to the plane spanned by <math>\vec{a}</math> and <math>\vec{b}</math> with the direction given by '''the right-hand rule''': If you put the '''[[Index finger|index]]''' of your right hand on <math>\vec{a}</math> and the [[middle finger]] on <math>\vec{b}</math>, then the [[thumb]] points in the direction of <math>\vec{a}\times\vec{b}</math>.<ref>{{Cite book |last=Hubbard |first=John H. (John Hamal) |url=http://archive.org/details/vectorcalculusli0000hubb |title=Vector calculus, linear algebra, and differential forms : a unified approach |date=2009 |publisher=Ithaca, NY : Matrix Editions |others=Internet Archive |isbn=978-0-9715766-5-0}}</ref> [[File:Fleming's right hand rule.png|left|thumb|Fleming's right hand rule]] The right-hand rule in physics was introduced in the late 19th century by [[John Ambrose Fleming|John Fleming]] in his book Magnets and Electric Currents.<ref name=":0">{{Cite book |last=Fleming |first=J. A. (John Ambrose) |url=http://archive.org/details/magnetsandelect01flemgoog |title=Magnets and electric currents. An elementary treatise for the use of electrical artisans and science teachers |date=1902 |publisher=London, E. & F.N. Spon, limited; New York, Spon & Chamberlain |others=Harvard University}}</ref> Fleming described the orientation of the induced electromotive force by referencing the motion of the conductor and the direction of the magnetic field in the following depiction: “If a conductor, represented by the middle finger, be moved in a field of [[magnetic flux]], the direction of which is represented by the direction of the [[Index finger|forefinger]], the direction of this motion, being in the direction of the thumb, then the electromotive force set up in it will be indicated by the direction in which the middle finger points."<ref name=":0" />
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