6. To reduce a given bivector
to the above form, we may set
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where
and
are scalars, which we may regard as known. The value of
may be determined by the equation
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the quadrant to which
belongs being determined so as to give
and
the same signs as
and
Then
and
will be given by the equation
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The solution ia indeterminate when the real and imaginary parts of the given bivector are perpendicular and equal in magnitude. In this case the directional ellipse is a circle, and the bivector may be called circular. The criterion of a circular bivector is
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It is eepecially to be noticed that from this equation we cannot conclude that
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as in the analysis of real vectors. This may also be shown by expressing
in the form
in which
are biacalars. The equation then becomes
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which evidently does not require
and
to vanish, as would be the case if only real values are considered.
7. Def.—We call two vectors
and
perpendicular when
Allowing the same analogy, we shall call two bivectors
and
perpendicular, when
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In considering the geometrical signification of this equation, we shall first suppose that the real and imaginary components of
and
lie in the same plane, and that both
and
have not real directions. It is then evidently possible to expreas them in the form
![{\displaystyle m[\alpha +\iota \beta ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b217cd686ee003b61c417678779c248abf11f90a)
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where
and
are biscalar,
and
are at right angles, and
parallel with
Then the equation
requires that
and
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This shows that the directional ellipses of the two bivectors are similar and the angular direction from the real to the imaginary