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
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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