as will be evident on considering separately in the expression
the components perpendicular and parallel to
or on substituting in
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for
and
their values in terms of
If we set, in either of these equations,
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we obtain, on reduction, the formula
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in which the versor is expressed in terms of the rectangular components of the vector semitangent of version.
146. If
are unit vectors, expressions of the form
![{\displaystyle 2\alpha \alpha -{\text{I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23ff2f92a857668b6f7808102de4454e63dff266) ![{\displaystyle 2\beta \beta -{\text{I}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa849f147bedef495594149a32d1b21ae2e0b93a)
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are biquadrantal versors. A product like
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is a versor of which the axis is perpendicular to
and
and the amount of rotation twice that which would carry
to
It is evident that any versor may be thus expressed, and that either
or
may be given any direction perpendicular to the axis of rotation. If
and
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we have for the resultant of the successive rotations
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This may be applied to the composition of any two successive rotations,
being taken perpendicular to the two axes of rotation, and affords the means of determining the resultant rotation by construction on the surface of a sphere. It also furnishes a simple method of finding the relations of the vector semitangents of version for the versors
and
Let
![{\displaystyle \theta _{1}={\frac {-\Phi _{\text{X}}}{1+\Phi _{\text{S}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc4206e204592548abeddede135209872fab498) ![{\displaystyle \theta _{2}={\frac {-\Psi _{\text{X}}}{1+\Psi _{\text{S}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf88f2645884191dcb22facf7890d79edf7e855)
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Then, since
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which is moreover geometrically evident. In like manner,
![{\displaystyle \theta _{2}={\frac {\beta \times \gamma }{\beta .\gamma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e56d6a54db0bf77237ce36cfbddf81c4a05a270)
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