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

60 If $$\alpha$$ is a unit vector, If $$i, j, k$$ are a normal system of unit vectors  If $$\alpha$$ and $$\beta$$ are any vectors,  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. This is essentially the theorem of No. 27, expressed in a form more symmetrical, and more easily remembered.

132. The equation gives, on multiplication by any vector $$\rho,$$ the identical equation  (See No. 37.) The former equation is therefore identically true. (See No. 108.) It is a little more general than the equation 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 Multiplying this equation by $$\rho$$ as prefactor (or the first equation by $$\rho$$ as postfactor), we obtain  (Compare No. 37.) For three complanar vectors we have  Multiplying this by $$\nu,$$ a unit normal to the plane of $$\alpha, \beta,$$ and $$\gamma$$ we have