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Othogonality meets Quaternions
Name: benjamin p heroux
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
I have been told many a time that imaginary and real numbers were othogonal.
What third othogonal component would complete a description of three
dimensions? A part of quaternions? If so, is there a simple algorithm
for figuring the properties of hyperquaternions, hyperhyperquaternions,
etc.?
Replies:
Yes, quaternion algebra does make a somewhat neat representation of cross
and dot products in 3-dimensional space, just as complex algebra can
represent some aspects of 2 dimensions very well. The way it works for
quaternions is that a point x,y,z in space is represented by the quaternion
q = x i + y j + z k
and then q1 * q2 in quaternion algebra (where i^2 = -1, i*j = k etc)
gives
q1 * q2 = -(x1*x2 + y1*y2 + z1*z2) + (y1 z2 - z1 y2) i +
(z1 x2 - x1 z2) j + (x1 y2 - y1 x2) k
= - D + x3 i + y3 j + z3 k
where D is the 3-dimensional dot product of (x1,y1,z1) and (x2,y2,z2)
and (x3,y3,z3) is the 3-dimensional cross product.
However, I have never seen this used anywhere, or found to be particularly
useful. One problem with quaternion algebra, unlike complex algebra is that
it is non-commutative: q1 * q2 is not the same as q2 * q1. Any extension to
hyper-quaternions etc. makes things even worse: the results are forced to
be non-associative and therefore lose even the general "group" properties (I
believe they are called "loops").
asmith
Nice explanation, asmith!
Just for your information, quaternions are used in computational chemistry
to enforce rigid-body constraints on classical dynamics simulations of
certain large molecules . . . so there is an application you may not have
seen.
However, it should be noted that there are more numerically efficient
methods than quaternion methods (the SHAKE algorithm comes to mind) and so
only a small fraction of compu-chemists use quaternions for this purpose.
rtopper
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Update: June 2012
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