I know that almost anyone knows what's a vector. Some say that is an arrow. Others that is an object that has both a length and direction. Well, let's examine that in depth.
First of all, a vector is a logical definition from the Set Theory.
Let R be an equivalence relation for a set K. We define the Quotient Set, denoted by the symbol
, as:
![K/R := \{ [X]_R | X \in K \}](http://felipetonello.com/blog/wp-content/plugins/latex/cache/tex_baf740b15b597c53b2b9acac1b7e3b0d.gif)
Results that
is a partition of K, i.e.,
is a class of non-empty subsets of K, pairwise disjoint and whose union is the class K.
In summary, we have the following important result:
If R an equivalence relation on a set K, than, the quotient set of K for R is a partition of K.
R is a relation defined in class K from all oriented segments in the plane, as follows:
if, and only if, the length of xy is same as uz, the line which support xy is parallel to uz and xy has the same direction as uz.
Results that R is an equivalence in K. Therefore:
is a partition of K. So,
![PQ \ R\ P'Q' \leftrightarrow [PQ]_R = [P'Q']_R](http://felipetonello.com/blog/wp-content/plugins/latex/cache/tex_40f444cf93e82129f1b7da0f46279803.gif)
Remembering that
. If you want to know more about, check the Equivalence Class.
So, a vector from the plane is, by definition, any one element of the quotient set
. Thus we have,

If you think this is too much. There is an easy approach.
Vector is a set of all equipolent oriented segments together. i.e.

You have never wondered why it is correct to use equal sign (
) for "different but similar" vectors and it is incorrect to use equal sign for "different but similar" oriented segments? The truth is, since you are comparing sets(vectors) with the same elements(oriented segments), it is true when you say
. Now, if
and
were oriented segments, you will only be able to use the equal sign if a and b are really the same element in the plane(the same set of points).
Yes, logics is much more than computer algorithm or stuff like that.
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