Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/38

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22

VECTOR ANALYSIS.

and, in general, direct and skew products of sums of vectors may be expanded precisely as the products of sums in algebra, except that in skew products the order of the factors must not be changed without compensation in the sign of the term. If any of the terms in the factors have negative signs, the signs of the expanded product (when there is no change in the order of the factors) will be determined by the same rules as in algebra. It is on account of this analogy with algebraic products that these functions of vectors are called products and that other terms relating to multiplication are applied to them.

21. Numerical calculation of direct and skew products.—The properties demonstrated in the last two paragraphs (which may be briefly expressed by saying that the operations of direct and skew multiplication are distributive) afford the rule for the numerical calculation of a direct product, or of the components of a skew product, when the rectangular components of the factors are given numerically. In fact, if

\alpha =xi+yj+zk,

⁠and⁠

\beta =x'i+y'j+z'k;

\alpha .\beta =xx'+yy'+zz',

and

\alpha \times \beta =(yz'-zy')i+(zx'-xz')j+(xy'-yx')k.

22. Representation of the area of a parallelogram by a skew product.—It will be easily seen that $\alpha \times \beta$ represents in magnitude the area of the parallelogram of which $\alpha$ and $\beta$ (supposed drawn from a common origin) are the sides, and that it represents in direction the normal to the plane of the parallelogram on the side on which the rotation from $\alpha$ toward $\beta$ appears coiuter-clockwise.

23. Representation of the volume of a parallelopiped by a triple product.—It will also be seen that $\alpha \times \beta .\gamma$ ^[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-clockwise, or on the opposite side.

24. Hence,

{\begin{aligned}\alpha \times \beta .\gamma &=\gamma \times \alpha .\beta =\gamma .\alpha \times \beta =\alpha .\beta \times \gamma \\&=\beta .\gamma \times \alpha =-\beta \times \alpha .\gamma =-\gamma \times \beta .\alpha =-\alpha \times \gamma .\beta \\&=-\gamma .\beta \times \alpha =-\alpha .\gamma \times \beta =-\beta .\alpha \times \gamma .\end{aligned}}

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

↑ 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 $[\alpha \times \beta ].\gamma$ .

[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 $[\alpha \times \beta ].\gamma$ .

[1]