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Schwere, Elektricität und Magnetismus:377

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Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.


Direct and Skew Products of Vectors.


 13. Def.—The direct product of and (written ) is the scalar quantity obtained by multiplying the product of their magnitudes, by the cosine of the angle made by their directions.

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

 The direction of may also be defined as that in which an ordinary screw advances as it turns so as to carry toward .

 Again, if be directed toward the east, and lie in the same horizontal plane and on the north side of , will be directed upward.

 15. It is evident from the preceding definitions that


and


 16. Moreover,



and


The brackets may therefore be omitted in such expressions.

 17. From the definitions of No. 11 it appears that







 18. If we resolve into two components and , of which the first is parallel and the second perpendicular to , we shall have


and


 19. and .

 To prove this, let , and resolve each of the vectors into two components, one parallel and the other perpendicular to . Let these be . Then the equations to be proved will reduce by the last section to


and