Schwere, Elektricität und Magnetismus:379

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

ogram on the side on which the rotation from $\alpha \!$ toward $\beta \!$ appears counter-clock-wise.

 23. Representation of the volume of a parallelopiped by a triple product.—It will also be seen that $\alpha \times \beta \cdot \gamma \!$*^[1] represents in numerical value the volume of the parallelopiped of which $\alpha \,$, $\beta \,$ and $\gamma \!$ (supposed drawn from a common origin), are the edges, and that the value of the expression is positive or negative according as $\gamma \!$ lies on the side of the plane of $\alpha \!$ and $\beta \!$ on which the rotation from $\alpha \!$ to $\beta \!$ appears counter-clock-wise, or on the opposite side.

 24. Hence,

$\alpha \times \beta \cdot \gamma =\beta \times \gamma \cdot \alpha =\gamma \times \alpha \cdot \beta =\gamma \cdot \alpha \times \beta =\alpha \cdot \beta \times \gamma \,$ $=\beta \cdot \gamma \times \alpha =-\beta \times \alpha \cdot \gamma =-\gamma \times \beta \cdot \alpha =-\alpha \times \gamma \cdot \beta \,$ $=-\gamma \cdot \beta \times \alpha =-\alpha \cdot \gamma \times \beta =-\beta \cdot \alpha \times \gamma .\,$

It will be observed that all the products of this type, which can be made with three given vectors, are the same in numerical value, and that any two such products are of the same or opposite character in respect to sign, according as the cyclic order of the letters is the same or different. The product vanishes when two of the vectors are parallel to the same line, or when the three are parallel to the same plane.

 This kind of product may be called the scalar product of the three vectors. There are two other kinds of products of three vectors, both of which are vectors, viz: products of the type $(\alpha \cdot \beta )\gamma \!$ or $\gamma (\alpha \cdot \beta )\!$, and products of the type $\alpha \times \lbrack \beta \times \gamma \rbrack \!$ or $\lbrack \gamma \times \beta \rbrack \times \alpha \!$.

25.	$i\cdot j\times k=j\cdot k\times i=k\cdot i\times j=1.\quad i\cdot k\times j=k\cdot j\times i=j\cdot i\times k=-1.\,$

From these equations, which follow immediately from those of No. 17, the propositions of the last section might have been derived, viz: by substituting for $\alpha \,$, $\beta \,$, and $\gamma \!$ respectively, expressions of the form $xi+yj+zk,\,x'i+y'j+z'k\!$, and $x''i+y''j+z''k\!$.†^[2] Such a method, which may be called expansion in terms of $i,\,j$ and $k\!$, will on many occasions afford very simple, although perhaps lengthy, demonstrations.

 26. Triple products containing only two different letters. — The significance and the relations of $(\alpha .\alpha )\beta \!$ and $\alpha \times \lbrack \alpha \times \beta \rbrack \!$ will be most evident, if we consider $\beta \!$ as made up of

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1. * Since the sign $\times \!$ is only used between vectors, the skew multiplication in expressions of this kind is evidently to be performed first. In other words, the above expression must be interpreted as $\lbrack \alpha \times \beta \rbrack \cdot \gamma \!$.
2. † The student who is familiar with the nature of determinants will not fail to observe that the triple product $\alpha \cdot \beta \times \gamma \!$ is the determinant formed by the nine rectangular components of $\alpha \,$, $\beta \,$ and $\gamma \!$, nor that the rectangular components of $\alpha \times \beta \!$ are determinants of the second order formed from the components of $\alpha \!$ and $\beta \!$. (See the last equation of No. 21.)