Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Othogonality meets Quaternions
Name: benjamin p heroux
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995

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.?

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)

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").


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.


Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory