the
, and
components of
and adding. Hence the second equation may be regarded as the most general form of a scalar equation of the first degree in
etc., which can be derived from the original vector equation or its equivalent three scalar equations. If we wish to have two of the scalars, as
and
, disappear, we have only to choose for
a vector perpendicular to
and
. Such a vector is
. We thus obtain
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37. Relations of four vectors.—By this method of elimination we may find the values of the coefficients
, and
in the equation
|
(1)
|
by which any vector
is expressed in terms of three others. (See No. 10.) If we multiply directly by
, and
, we obtain
![{\displaystyle \rho .\beta \times \gamma =a\alpha .\beta \times \gamma ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63aaa985b3674637c42f8bd401f9a4cdf4ffd3f7) ![{\displaystyle \rho .\gamma \times \alpha =b\beta .\gamma \times \alpha ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d457eca2332354cf391fb4708e6ceaccbb3f2cc)
|
(2)
|
whence
![{\displaystyle a={\frac {\rho .\beta \times \gamma }{\alpha .\beta \times \gamma }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1919e5a9e117897ebde85efdb4144912307fb61d) ![{\displaystyle b={\frac {\rho .\gamma \times \alpha }{\alpha .\beta \times \gamma }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddacf620d7ec3d4dbb58a9465ce8b7539d5e90fa)
|
(3)
|
By substitution of these values, we obtain the identical equation,
|
(4)
|
(Compare No. 31.) If we wish the four vectors to appear symmetrically in the equation we may write
|
(5)
|
If we wish to express
as a sum of vectors having directions perpendicular to the planes of
and
of
and
, and of
and
, we may write
|
(6)
|
To obtain the values of
, we multiply directly by
, by
, and by
. This gives
![{\displaystyle e={\frac {\rho .\alpha }{\beta .\gamma \times \alpha }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78e824a0c2c529f37bdc82664b10b85141fde811) ![{\displaystyle f={\frac {\rho .\beta }{\gamma .\alpha \times \beta }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a548f87f62031e87465227336dc21c6141e8f2d7)
|
(7)
|
Substituting these values we obtain the identical equation
|
(8)
|
(Compare No. 32.)
38. Reciprocal systems of vectors.— The results of the preceding section may be more compactly expressed if we use the abbreviations
![{\displaystyle \alpha '={\frac {\beta \times \gamma }{\alpha .\beta \times \gamma }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e786f5f4e05acc0f12011c3d768e7e6fb3e0681a)
|
(1)
|
The identical equations (4) and (8) of the preceding number thus become
|
(2)
|
|
(3)
|