ns am currently learning calculus and also am tho a little confused as to the difference in between vectors and points (which are represented as ordered pairs). I recognize that vectors room a different form of object offered that they have both direction and also magnitude, but I don"t understand why they are inherently various given the they perform not it seems to be ~ to connect any much more information 보다 a suggest represented as an bespeak pair does.

Furthermore, is over there something "special" I have to do to convert an bespeak pair $(a,b)$ into a vector $$ or not?


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inquiry Feb 13 "18 in ~ 21:38
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mfarringtonmfarrington
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I understand your confusion and I"m going to try help you a bit.

You are watching: Describe the vector as an ordered pair

First: Vectors room the aspects of a vector space. Ns don"t know if you know the (abstract) definition of a vector space. Anyway, vectors are much much more than points or arrows. Because that example, real numbers can be considered themselves together a vectors. And also the same use for bag $(x,y)$, triples $(x,y,z)$ and so on. Continous functions defined on an interval $$ is another example that vector space; here the vectors room the functions. So, together you can see, vectors space a really rich topic.

Second: If her vector an are is actual (complex) and also finite-dimensional then, you can study it as the vector room $chathamtownfc.netbb R^n$ (resp. $chathamtownfc.netbb C^n$), for some appropiate $n$. So, at the end of the day, you have actually points.

Third: most likely your man cames from the framework of affine spaces. Around speaking, an affine room is a vector an are $V$ in addition to a pair $(P,g)$, wherein $P$ is a collection (the collection of points) and $g:P imes P ightarrow V$ is a map the assigns a vector to any pair of points $p,q$, (with some rules).

Now, ~ above this situation, imagine yo have actually a preferred suggest $chathamtownfc.netcal O$ ~ above $P$, which we will speak to the origin. Then, for every point $pin P$, friend have characterized a canonical vector ~ above $V$, specific the vector

$$ g(chathamtownfc.netcal O, p) equiv vecchathamtownfc.netcal O pequiv vec ns . $$

This vector has an application suggest ($chathamtownfc.netcal O$), direction and length (magnitude) and also is various than the suggest $p$.

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So, answering one of your questions yes, for any type of pair of point out you have a means to specify a vector: the map $g$.