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
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16. Moreover,
and
The brackets may therefore be omitted in such expressions.
17. From the definitions of No. 11 it appears that
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18. If we resolve into two components and , of which the first is parallel and the second perpendicular to , we shall have
and
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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
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Now since we may form a triangle in space, the sides of which shall be , and Projecting this on a plane perpendicular to , we obtain a triangle having the sides and which affords the relation If we pass planes perpendicular to a through the vertices of the first triangle, they will give on a line parallel to a segments equal to Thus we obtain the relation Therefore since all the cosines involved in these products are equal to unity. Moreover, if is a unit vector, we shall evidently have since the effect of the skew multiplication by a upon vectors in a plane perpendicular to a is simply to rotate them all 90° in that plane. But any case may be reduced to this by dividing both sides of the equation to be proved by the magnitude of The propositions are therefore proved.
20. Hence,
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