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

mfarringtonmfarrington
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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\$.