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
|