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Re: [Phys-l] bound vectors ... or not



> Eh? Based on the wiki article you cited, a bound vector "possesses a
definite initial point and terminal point." By this simple reading,
both r and F qualify.

I disagree. This seems to be exactly the fallacy I alluded
to a moment ago.

As I see it, a vector has, by definition, a direction and
a magnitude. A vector in 3-space can be specified by
three numbers. If some funny kind of vector actually
had a definite starting-point and a definite ending-point,
we would need six numbers to specify it.

By using only 3 numbers, is it not implied that such a vector's tail starts at the origin, and hence is specified by 6 numbers? Regardless, I'm not disagreeing with you per se - see next paragraph.

Looked at another way, we all know how to do vector addition - and in doing so correctly, we know that the tail of one vector has to be placed at the head of another for the resultant to be correctly determined. This would imply that both the placement of the head and tail (6 numbers), in some coordinate space, are necessary. At the same time, it's my position (not to pun) that since a vector is a magnitude and direction, once you place the tail (or the head) alone, the rest is determined, hence really only 3 numbers are required, so I'm on your page in the end.

Anyway, the situation is worse than we thought. In consulting a 1959 text on engineering mechanics, by J Meriam (no not J Marion of physics fame), there are not 1, not 2, but 3 vectors: free, sliding, and fixed (the latter of which is what is being called bound here). Frankly, I don't feel like paraphrasing or quoting his descriptions - it doesn't seem wrong for teaching mechanics as espoused in his text, just that it seems unnecessarily complicated.


Torque is a bivector. Angular momentum is a bivector.

To play devil's advocate, I never see the term bivector used either. Just like the term "bound vector." At their most explicit, T&L are referred to as vector cross products, but are further referred to as just another vector. As such, one can again then add L1 + L2 etc using the same old rules of vector addition etc. In the end, I suspect you might often get responses like "what is the purpose of this bivector formalism, with its extensions? I don't need it for my classical mechanics or classical E&M."


Question: on your clifford page and its discussion of bivectors, why do you (apparently) maintain what I would call a conventional view of the dot product but eschew the term "cross product" in lieu of wedge product (if I read it correctly)? It appears to be only because of the (not unimportant) extension to higher dimensions (that is, all cross products are wedge, but not all wedge are cross) not to mention symmetry considerations. Fine, but I think the engineering proponents of free and bound vectors are firmly grounded in 3 dimensions at most. The discussion of bivector, for them, is an unnecessary complication, just as the distinction between free and bound has also been in this discussion.


Stefan Jeglinski