The criterion of a versor may therefore be written
and
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For the last equation we may substitute
or
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It is evident that the resultant of successive finite rotations is obtained by multiplication of the versors.
143. If we take the axis of the rotation for the direction of
will have the same direction, and the versor reduces to the form
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in which
and
are normal systems of unit vectors.
We may set
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and the versor reduces to
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or
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where
is the angle of rotation, measured from
toward
if the versor is used as a prefactor.
144. When any versor
is used as a pref actor, the vector
will be parallel to the axis of rotation, and equal in magnitude to twice the sine of the angle of rotation measured counter-clockwise as seen from the direction in which the vector points. (This will appear if we suppose
to be represented in the form given in the last paragraph.) The scalar
will be equal to unity increased by twice the cosine of the same angle. Together,
and
determine the versor without ambiguity. If we set
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the magnitude of
will be
or
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where
is measured counter-clockwise as seen from the direction in which
points. This vector
which we may call the vector semitangent of version, determines the versor without ambiguity.
145. The versor
may be expressed in terms of
in various ways. Since
(as pref actor) changes
into
(
being any vector), we have
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Again
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