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One can specify a rotation in n dimensions by specifying two ''reflections''. Indeed, this is the definition of a rotation. Each reflection can then in turn be specified by a corresponding unit vector, orthogonal to the (n-1) dimensional subspace which is invariant under the reflection. There are two such vectors: '''A''' and '''-A'''. If we choose positive notation by convention, '''A''' is mapped to '''-A''' under the reflection represented by '''A'''.
Composing with a second reflection '''B''' moves '''-A''' to its final desination = 2['''A'''|'''B'''] '''B''' - '''A''' , where [x|y] specifies the finite dimensional inner product, following the notation of quantum mechanics for infinite dimensional inner products (<x|y>). ['''A'''|'''B'''] is of course the cosine of the angle between '''A''' and '''B''', call it ''t'', and defines this angle unambiguously. Therefore, '''A''' is rotated by an angle 2''t ''through '''B''', in the two dimensional subspace containing both '''A''' and '''B'''. The 'axis' of rotation is the n-2 dimensional subspace fixed by the composition of the two reflections '''A''' and '''B''' and is orthogonal to the plane containing '''A''' and '''B'''. I say 'axis' because its use in this context is an unforgivable corruption of terminology.
Returning to the topic at hand, namely quaternions, one may form the rotation which takes the three dimensional vector '''A''' to '''B''' in the plane of '''A''' and '''B''' by calculating sqrt('''AB'''). As the second contributor correctly notes, there are infinitely many other rotations which move '''A''' to '''B'''. Of course, if '''A''' and '''B''' are not of unit magnitude, the resulting square root needs to be normalized.
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