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Re: why are there [v e c t o r s] in Lisp?


From: Pascal J. Bourguignon
Subject: Re: why are there [v e c t o r s] in Lisp?
Date: Sat, 17 Oct 2015 04:22:58 +0200
User-agent: Gnus/5.13 (Gnus v5.13) Emacs/24.3 (gnu/linux)

Aurélien Aptel <aurelien.aptel+emacs@gmail.com> writes:

> I'm just nitpicking, but:
> […]
> In pure linear algebra, "direction" and "position" are not defined for
> vectors. 

Well once you define dot-product, you have defined implicitely angles
and therefore direction.

    A⋅B = |A|×|B|×cos(∠AB)
    A⋅B/(|A|×|B|) = cos(∠AB)

    For a vector V and a base (In), 
    (cos(∠VIn))=(V⋅In/|V|) defines the direction of the vector V.

    If (In)=((δin)), then (V⋅In/|V|) = V/|V|
    which shows that any vector defines a direction by itself.

The essence of vector is to define a direction and a magnitude.  Notably
for vector spaces without a finite basis, V/|V| is still the direction
of the vector V, intrinsically (every unitary vector is a distinct
direction).

-- 
__Pascal Bourguignon__                 http://www.informatimago.com/
“The factory of the future will have only two employees, a man and a
dog. The man will be there to feed the dog. The dog will be there to
keep the man from touching the equipment.” -- Carl Bass CEO Autodesk


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