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
<[log in to unmask]> 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
>> <[log in to unmask]> 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
>>>> [log in to unmask]
>>>
>>>
>>>
>>>
>
>
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