Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/35

When the vectors to be 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 $\alpha +[\beta +\gamma ]+{\text{etc.,}}$ 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 conmiutative 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.,

${\begin{aligned}(m+n+{\text{etc.}})[\alpha +\beta +{\text{etc.}}]&=m\alpha +n\beta &+{\text{etc.}}\\&+m\beta +n\beta &+{\text{etc.}}\\&&+{\text{etc.}}\end{aligned}}$

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 multiplications may be applied to the vector equations, since the coefficients are scalars.

10. Linear relation of four vectors, Coordinates.—If $\alpha ,\beta$, and $\gamma$ are any given vectors not parallel to the same plane, any other vector $\rho$ may be expressed in the form

$\rho =a\alpha +b\beta +c\gamma .$

If $\alpha ,\beta$, and $\gamma$ are unit vectors, $a,b$, and $c$ are the ordinary scalar components of $\rho$ parallel to $\alpha ,\beta$, and $\gamma$. If $\rho ={\overline {\text{OP}}}$, ($\alpha ,\beta ,\gamma$ being unit vectors), $a,b$, and $c$ are the cartesian coordinates of the point ${\text{P}}$ referred to axes through ${\text{O}}$ parallel to $\alpha ,\beta$, and $\gamma$. When the values of these scalars are given, $\rho$ is said to be given in terms of $\alpha ,\beta$, and $\gamma$.