Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/76

If $\alpha$ is a unit vector,

${\begin{aligned}\{{\text{I }}\times \alpha \}^{2}&=-\{{\text{I}}-\alpha \alpha \},\\\{{\text{I }}\times \alpha \}^{3}&=-{\text{I }}\times \alpha ,\\\{{\text{I }}\times \alpha \}^{4}&={\text{I}}-\alpha \alpha ,\\\{{\text{I }}\times \alpha \}^{5}&={\text{I }}\times \alpha ,\\{\text{etc.}}\end{aligned}}$

If $i,j,k$ are a normal system of unit vectors

${\begin{aligned}{\text{I }}\times {\text{ I}}&={\text{I }}\times {\text{ I}}=kj-jk,\\{\text{I }}\times j&=j\times {\text{ I}}=ik-ki,\\{\text{I }}\times k&=k\times {\text{ I}}=ji-ij.\end{aligned}}$

If $\alpha$ and $\beta$ are any vectors,

$[\alpha \times \beta ]\times {\text{ I}}={\text{I }}\times [\alpha \times \beta ]=\beta \alpha -\alpha \beta .$

That is, the vector $\alpha \times \beta$ as a pre- or post-factor in skew multiplication is equivalent to the dyadic $\{\beta \alpha -\alpha \beta \}$ taken as pre- or postfactor in direct multiplication.

${\begin{aligned}[\alpha \times \beta ]\times \rho &=\{\beta \alpha -\alpha \beta \}.\rho ,\\\rho \times [\alpha \times \beta ]&=\rho .\{\beta \alpha -\alpha \beta \}.\end{aligned}}$

This is essentially the theorem of No. 27, expressed in a form more symmetrical, and more easily remembered.

132. The equation

$\alpha \,\beta \times \gamma +\beta \,\gamma \times \alpha +\gamma \,\alpha \times \beta =\alpha .\beta \times \gamma \,{\text{I}}$

gives, on multiplication by any vector $\rho ,$ the identical equation

$\rho .\alpha \,\beta \times \gamma +\rho .\beta \,\gamma \times \alpha +\rho .\gamma \,\alpha \times \beta =\alpha .\beta \times \gamma \,\rho .$

(See No. 37.) The former equation is therefore identically true. (See No. 108.) It is a little more general than the equation

$\alpha \alpha '+\beta \beta '\gamma \gamma '={\text{I}},$

which we have already considered (No. 124), since, in the form here given, it is not necessary that $\alpha ,\beta ,$ and $\gamma$ should be non-complanar. We may also write

$\beta \times \gamma \,\alpha +\rho .\gamma \times \alpha \,\beta +\alpha \times \beta \,\gamma =\alpha .\beta \times \gamma \,{\text{I}}.$

Multiplying this equation by $\rho$ as prefactor (or the first equation by $\rho$ as postfactor), we obtain

$\rho .\beta \times \gamma \,\alpha +\rho .\gamma \times \alpha \,\beta +\rho .\alpha \times \beta \,\gamma =\alpha .\beta \times \gamma \,\rho .$

(Compare No. 37.) For three complanar vectors we have

$\alpha \,\beta \times \gamma +\beta \,\gamma \times \alpha +\gamma \,\alpha \times \beta =0.$

Multiplying this by $\nu ,$ a unit normal to the plane of $\alpha ,\beta ,$ and $\gamma$ we have

$\alpha \,\beta \times \gamma .\nu +\beta \,\gamma \times \alpha .\nu +\gamma \,\alpha \times \beta .\nu =0.$