Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/39

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 $(\alpha .\beta )\gamma$ or $\gamma (\alpha .\beta )$, and products of the type $\alpha \times [\beta \times \gamma ]$ or $[\gamma \times \beta ]\times \alpha$.

25. $i.j\times k=j.k\times i=k.i\times j=1.$⁠$i.k\times j=k.j\times i=j.i\times k=-1.$

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 $x''i+y''j+z''k$.^[1] Such a method, which may be called expansion of 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 $(\alpha .\alpha )\beta ,(\alpha .\beta )\alpha$, and $\alpha \times [\alpha \times \beta ]$ will be most evident, if we consider $\beta$ as made up of two components, $\beta '$ and $\beta ''$, respectively parallel and perpendicular to $\alpha$. Then

	$\beta$	$=\beta '+\beta '',$
	$(\alpha .\beta )\alpha$	$=(\alpha .\beta ')\alpha =(\alpha .\alpha )\beta ',$
	$\alpha \times [\alpha \times \beta ]$	$=\alpha \times [\alpha \times \beta '']=-(\alpha .\alpha )\beta ''.$
Hence, ⁠	$\alpha \times [\alpha \times \beta ]$	$=(\alpha .\beta )\alpha -(\alpha .\alpha )\beta .$

27. General relation of the vector prodAicts of three factors.—In the triple product $\alpha \times [\beta \times \gamma ]$ we may set

$\alpha =l\beta +m\gamma +n\beta \times \gamma ,$

unless $\beta$ and $\gamma$ have the same direction. Then

${\begin{aligned}\alpha \times [\beta \times \gamma ]&=l\beta \times [\beta \times \gamma ]+m\gamma \times [\beta \times \gamma ]\\&=l(\beta .\gamma )-l(\beta .\beta )\gamma -m(\gamma .\beta )\gamma +m(\gamma .\gamma )\beta \\&=(l\beta .\gamma +m\gamma .\gamma )\beta -(l\beta .\beta +m\gamma .\beta )\gamma .\end{aligned}}$

But	$l\beta .\gamma +m\gamma .\gamma =\alpha .\gamma$,⁠and⁠$l\beta .\beta +m\gamma .\beta =\alpha .\beta .$

Therefore

$\alpha \times [\beta \times \gamma ]=(\alpha .\gamma )\beta -\gamma (\beta .\alpha ),$

which is evidently true, when $\beta$ and $\gamma$ have the same directions. It may also be written

$[\gamma \times \beta ]\times \alpha =\beta (\gamma .\alpha )-\gamma (\beta .\alpha ).$

1. The student who is familiar with the nature of determinants will not fail to observe that the triple product $\alpha .\beta \times \gamma$ 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.)