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Let $F(v) = qvq^{-1}$ be a mapping of points $v$ in $R^3$ to images in $R^3$ where the 4-dimensional quaternion $q$ acts as an operator. We have proved that this mapping fixes every point on the x axis. |
(2) In this section we consider the mapping $G(v) = qvq^{-1}$ of $R^3$ to $R^3$ where the quaternion $q = \cos \theta + \sin \theta {\bf k}$ is an operator. |
We have shown $qkq^{-1} = k$ so $G(v+tk)= q(v +tk)q^{-1} =
qvq^{-1} + qtkq^{-1} = qvq^{-1} + tk$ for all $t$.
We can see that the vector $v$ in the $xy$ plane is rotated
about the $z$ axis by an angle $2\theta$ and all points on the vertical line through
it are also rotated about the $z$-axis by an angle
$2\theta$.
So by the mapping $G(v) = qvq^{-1}$ all points in $R^3$ are
rotated by $2\theta$ about the $z$-axis.
Note that, for any rotation of $R^3$, we can make a
transformation of the coordinate system so that the axis of the
rotation is made to coincide with the $z$-axis, then perform the
rotation by the given angle about the $z$-axis, and finally
transform back to the original coordinate system.
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