Schwere, Elektricität und Magnetismus:377

VECTOR ANALYSIS.

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Direct and Skew Products of Vectors.

Vorlage:Idt213. Def.—The direct product of $\alpha$ and $\beta$ (written $\alpha \cdot \beta$) is the scalar quantity obtained by multiplying the product of their magnitudes, by the cosine of the angle made by their directions.

Vorlage:Idt214. Def.—The skew product of $\alpha$ and $\beta$ (written $\alpha \times \beta$) is a vector function of $\alpha$ and $\beta$. Its magnitude is obtained by multiplying the product of the magnitudes of $\alpha$ and $\beta$ by the sine of the angle made by their directions. Its direction is at right angles to $\alpha$ and $\beta$, and on that side of the plane containing $\alpha$ and $\beta$ (supposed drawn from a common origin), on which a rotation from $\alpha$ to $\beta$ through an arc of less than 180° appears countcr-clock-wise.

Vorlage:Idt2The direction of $\alpha \times \beta$ may also be defined as that in which an ordinary screw advances as it turns so as to carry $\alpha$ toward $\beta$.

Vorlage:Idt2Again, if $\alpha$ be directed toward the east, and $\beta$ lie in the same horizontal plane and on the north side of $\alpha$, $\alpha \times \beta$ will be directed upward.

Vorlage:Idt215. It is evident from the preceding definitions that

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Vorlage:Idt216. Moreover,

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The brackets may therefore be omitted in such expressions.

Vorlage:Idt217. From the definitions of No. 11 it appears that

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Vorlage:Idt218. If we resolve $\beta$ into two components ${\beta }^{\prime }$ and ${\beta }^{″}$, of which the first is parallel and the second perpendicular to $\alpha$, we shall have

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Vorlage:Idt219. $\alpha \cdot [\beta +\gamma ]=a\cdot \beta +\alpha \cdot \gamma$ and $\alpha \times [\beta +\gamma ]=\alpha \times \beta +\alpha \times \gamma$.

Vorlage:Idt2To prove this, let $\sigma =\beta +\gamma$, and resolve each of the vectors $\beta ,\gamma ,\sigma$ into two components, one parallel and the other perpendicular to $\alpha$. Let these be ${\beta }^{\prime },{\beta }^{″},{\gamma }^{\prime },{\gamma }^{″},{\sigma }^{\prime },{\sigma }^{″}$. Then the equations to be proved will reduce by the last section to

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