Any vector, whether in the 'mathematical' or 'physical' sense as defined in Wikipedia, and which is defined on a 3D vector space (Euclidean or otherwise - which I hope is what were talking about), has by definition 3 elements (real or complex). This clearly excludes all scalars (real or complex) which have only 1 whatever the dimension of the space. Therefore it's plainly impossible for an entity in 3D space to be both a scalar and a vector at the same time. Your conclusion that scalars and complex numbers fulfil the axioms of a vector space is applicable only in the case of a 1D vector space, and therefore is not relevant. My original observation which started this thread was intended to be general one, not for a particular special case.
-- Ian On Fri, Oct 15, 2010 at 5:17 PM, Douglas Theobald <dtheob...@brandeis.edu> wrote: > On Oct 15, 2010, at 11:37 AM, Ganesh Natrajan wrote: > >> Douglas, >> >> The elements of a 'vector space' are not 'vectors' in the physical >> sense. > > And there you make Ed's point -- some people are using the general vector > definition, others are using the more restricted Euclidean definition. > > The elements of a general vector space certainly can be physical, by any > normal sense of the term. And note that physical 3D space is not Euclidean, > in any case. > >> The correct Wikipedia page is this one >> >> http://en.wikipedia.org/wiki/Euclidean_vector >> >> >> Ganesh >> >> >> >> On Fri, 15 Oct 2010 11:20:04 -0400, Douglas Theobald >> <dtheob...@brandeis.edu> wrote: >>> As usual, the Omniscient Wikipedia does a pretty good job of giving >>> the standard mathematical definition of a "vector": >>> >>> http://en.wikipedia.org/wiki/Vector_space#Definition >>> >>> If the thing fulfills the axioms, it's a vector. Complex numbers do, >>> as well as scalars. >>> >>> On Oct 15, 2010, at 8:56 AM, David Schuller wrote: >>> >>>> On 10/14/10 11:22, Ed Pozharski wrote: >>>>> Again, definitions are a matter of choice.... >>>>> There is no "correct" definition of anything. >>>> >>>> Definitions are a matter of community choice, not personal choice; i.e. a >>>> matter of convention. If you come across a short squat animal with split >>>> hooves rooting through the mud and choose to define it as a "giraffe," you >>>> will find yourself ignored and cut off from the larger community which >>>> chooses to define it as a "pig." >>>> >>>> -- >>>> ======================================================================= >>>> All Things Serve the Beam >>>> ======================================================================= >>>> David J. Schuller >>>> modern man in a post-modern world >>>> MacCHESS, Cornell University >>>> schul...@cornell.edu >>> >>> >>> >>> > >
