Schwere, Elektricität und Magnetismus:375

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

Addition and Subtraction of Vectors.

 6. Def.—The sum of the vectors $\alpha ,\,\beta ,\And \!\!\!c.\,$ (written $\alpha +\beta +\!\!\!\And \!\!\!c.$) is the vector found by the following process. Assuming any point $A\!$, we determine successively the points $B,\,C,\And \!\!\!c.$, so that ${\overline {AB}}=\alpha ,{\overline {BC}}=\beta ,\And \!\!\!c.$ The vector drawn from $A\!$ to the last point thus determined is the sum required. This is sometimes called the geometrical sum, to distinguish it from an algebraic sum or an arithmetical sum. It is also called the resultant, and $\alpha ,\,\beta ,\And \!\!\!c.$, are called the components. When the vectors to he added are all parallel to the same straight line, geometrical addition reduces to algebraic: when they have all the same direction, geometrical addition like algebraic reduces to arithmetical.

 It may easily be shown that the value of a sum is not affected by changing the order of two consecutive terms, and therefore that it is not affected by any change in the order of the terms. Again, it is evident from the definition that the value of a sum is not altered by uniting any of its terms in brackets, as $\alpha +\lbrack \beta +\gamma \rbrack +\!\!\!\And \!\!\!c.\,$, which is in effect to substitute the sum of the terms enclosed for the terms themselves among the vectors to be added. In other words, the commutative and associative principles of arithmetical and algebraic addition hold true of geometrical addition.

 7. Def.—A vector is said to be subtracted when it is added after reversal of direction. This is indicated by the use of the sign $-\,$ instead of $+\,$.

 8. It is easily shown that the distributive principle of arithmetical and algebraic multiplication applies to the multiplication of sums of vectors by scalars or sums of scalars:—i. e.,

${\begin{alignedat}{2}(m+n+\!\!\!\And \!\!\!c.)\lbrack \alpha +\beta +\And \!\!\!c.\rbrack &=m\alpha &&+n\alpha +\!\!\!\And \!\!\!c.\,\\&+m\beta &&+n\beta +\!\!\!\And \!\!\!c.\,\\&\qquad &&\,+\!\!\!\And \!\!\!c.\,\end{alignedat}}$

 9. Vector Equations.—If we have equations between sums and differences of vectors, we may transpose terms in them, multiply or divide by any scalar, and add or subtract the equations, precisely as in the case of the equations of ordinary algebra. Hence, if we have several such equations containing known and unknown vectors, the processes of elimination and reduction by which the unknown vectors may be expressed in terms of the known are precisely the same, and subject to the same limitations, as if the letters representing vectors represented scalars. This will be evident if we consider that in the multiplications incident to elimination in the supposed scalar equations the multipliers are the coefficients of the unknown quantities, or functions of these coefficients, and that such