What is the physical origin of multiplying two 3-vectors

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One way quaternions can arise is by multiplying two vectors and assuming Hamilton's rule of i^2 = j^2 =k^2 = -1 (I think).

e.g. V = Vx*i + Vy*j + Vz*k, W = Wx*i + Wy*j + Wz*k

VW = (Vx*Wx + Vy*Wy + Vz*Wz)*(-1) + V x W = -V . W + V x W

Where x between two vectors is the cross product and a period between two vectors is the dot product.

My question is: what does multiplying two vectors in this way correspond to physcially? Adding two vectors could be, for example, finding the total distance from two distances in different directions. But what could the multiplication VW relate to physically?

-- Aaron Barlow (aaronb@ix.netcom.com), June 26, 2004


Hello Aaron:

The inability to answer this question was one reason quaternions were almost completely abandon by mathematical physicists. Hamilton promised they could do everything, but he did not deliver. His overselling of the project (and being Irish at a time when cultural racism was supported by nearly all), meant that quaternions don't appear in technical books today.

The key question is what does 1 quaternion represent. My answer: an event in spacetime. So a rock sitting on the sidewalk would represent a series of events that only changed in time, not in space. Toss that rock in the air, and the events that represent the trajectory would changing in x, y, and z (or whatever coordinate you choose).

I have taken two totally straight trajectories, quite dull as far as animations, and multiplied each subsequent moment of time together. The result was a curved line. I can't seem to find that animation, but other ones can be viewed at quaternions.com if you have the browser Opera with the mng plugin or using Konqueror on Linux.


-- Doug Sweetser (sweetser@alum.mit.edu), June 27, 2004.

lightning models

-- tarig sanhory (tarig90@hotmail.com), August 31, 2004.

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