Vectors

( A vector is something that mathematicians and engineers use to represent or calculate a force that has direction. The vector will show the direction of the force and the size of the force as well. Sometimes vectors are represented by arrows, with length or thickness of the arrow representing force, and direction representing direction. Sometimes vectors are represented by sets of numbers. Often vectors are used in spacecraft navigation. The engineers know where they want to go and use vectors to figure out what direction to thrust in to go there. A vector is an arrowed line whose length is proportional to a certain vector quantity and whose direction indicated the direction of the quantity

=History of (Topic)= ( Cartesian geometry, introduced by Fermat and Descartes around 1636, had a very large influence on mathematics bringing algebraic methods into geometry. By the middle of the 19th Century however there was some dissatisfaction with these coordinate methods and people began to search for direct methods, i.e. methods of synthetic geometry which were coordinate free.

It is possible however to trace the beginning of the vector concept back to the beginning of the 19th Century with the work of Bolzano. In 1804 he published a work on the foundations of elementary geometry //Betrachtungen über einige Gegenstände der Elementargoemetrie//. Bolzano, in this book, considers points, lines and planes as undefined elements and defines operations on them. This is an important step in the axiomatisation of geometry and an early move towards the necessary abstraction for the concept of a linear space to arise.

The move away from coordinate geometry was mainly due to the work of Poncelet and Chasles who were the founders of synthetic geometry. The parallel development in analysis was to move from spaces of concrete objects such as sequence spaces towards abstract linear spaces. Instead of substitutions defined by matrices, abstract linear operators must be defined on abstract linear spaces.

In 1827 Möbius published //Der barycentrische Calcul// a geometrical book which studies transformations of lines and conics. The novel feature of this work is the introduction of barycentric coordinates. Given any triangle //ABC// then if weights //a//, //b// and //c//are placed at //A//, //B// and //C// respectively then a point //P//, the centre of gravity, is determined. Möbius showed that every point //P//in the plane is determined by the homogeneous coordinates [//a//,//b//,//c//], the weights required to be placed at //A//, //B// and //C// to give the centre of gravity at //P//. The importance here is that Möbius was considering directed quantities, an early appearence of vectors.

In 1837 Möbius published a book on statics in which he clearly states the idea of resolving a vector quantity along two specified axes.

Between these two works of Möbius, a geometrical work by Bellavitis was published in 1832 which also contains vector type quantities. His basic objects are line segments //AB// and he considers //AB// and //BA// as two distinct objects. He defines two line segments as 'equipollent' if they are equal and parallel, so, in modern notation, two line segments are equipollent if they represent the same vector. Bellavitis then defines the 'equipollent sum of line segments' and obtains an 'equipollent calculus' which is essentially a vector space.

In 1814 Argand had represented the complex numbers as points on the plane, that is as ordered pairs of real numbers. Hamilton represented the complex numbers as a two dimensional vector space over the reals although of course he did not use these general abstract terms. He presented these results in a paper to the Irish Academy in 1833. He spent the next 10 years of his life trying to define a multiplication on the 3-dimensional vector space over the reals. Hamilton 's, published in 1843, was an important example of a 4-dimensional vector space but, particularly with Tait 's work on quaternions published in 1873, there was to be some competition between vector and quaternion methods. )

=Application of (Topic)= (vectors can be use in mathematical equations. for example a scalar quantity has only magnitude and is completely specified by a number and a unit. example are mas (a stone has a mas of 2kg), volume(a bottle has a volume of 12oz), and frequency(House current has a frequency of 60 cycles /s. Symbols of scalar quantities are printed in italic type(m=mas, V=volume. scalar quantities of the same kind are added using ordinary arithmetic. )

=References= > > )
 * 1) Beiser, Arthur. //Schaum's Outline of Theory and Problems of Physical Science//. New York: McGraw-Hill, 1988. Print.
 * 1) ( "Abstract Linear Spaces." //MacTutor History of Mathematics//. Web. 09 Aug. 2011. .
 * 1) (Ref #3 - from book, library, journal)

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