Schwere, Elektricität und Magnetismus:376

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

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

 10. Linear relation of four vectors, Coördinates.—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 $\rho ={\overline {OP}}$, ($\alpha ,\,\beta ,\,\gamma$ being unit vectors,) $a\,$, $b\,$, and $c\,$ are the cartesian coördinates of the point $P\,$ referred to axes through $O\,$ 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 $\gamma \,$. It is generally in this way that the value of a vector is specified, viz., in terms of three known vectors. For such purposes of reference, a system of three mutually perpendicular vectors have certain evident advantages.

 11. Normal systems of unit vectors.—The letters $i,\,j,\,k$ 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 and determined in direction by them in a perfectly definite manner. We shall always suppose that $k\,$ is on the side of the $i-j$ plane on which a rotation from $i\,$ to $j\,$ (through one right angle) appears counter-clock-wise. In other words, the directions of $i\,$, $j\,$, and $k\,$ are to be so determined that if they be turned (remaining rigidly connected with each other) so that $i\,$ points to the east, and $j\,$ to the north, $k\,$ will point upward. When rectangular axes of $X\,$, $Y\,$, and $Z\,$ are employed, their directions will be conformed to a similar condition, and $i,\,j,\,k$ (when the contrary is not stated) will be supposed parallel to these axes respectively. We may have occasion to use more than one such system of unit vectors, just as we may use more than one system of coördinate axes. In such cases, the different systems may be distinguished by accents or otherwise.

 12. Numerical computation of a geometrical sum.—If

${\begin{aligned}&\rho =a\,\alpha +b\,\beta +c\,\gamma ,\,\\\,\\&\sigma =a'\,\alpha +b'\,\beta +c'\,\gamma ,\,\\\,\\&\!\!\!\And \!\!\!c.,\,\end{aligned}}$

then

$\rho +\sigma +\And \!\!\!c.=(a+a'+\,\And \!\!\!c.)\alpha +(b+b'+\And \!\!\!c.)\beta +(c+c'+\And \!\!\!c.)\gamma .\,$

I. e., the coefficients by which a geometrical sum is expressed in terms of three vectors are the sums of the coefficients by which the separate terms of the geometrical sum are expressed in terms of the same three vectors.