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Topic: difference between a vector and a point (in R^n)
Replies: 12   Last Post: Sep 2, 2013 7:41 PM

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 Ken.Pledger@vuw.ac.nz Posts: 1,332 Registered: 12/3/04
Re: difference between a vector and a point (in R^n)
Posted: Aug 26, 2013 4:38 PM

lite.on.beta@gmail.com wrote:

> What is the difference between a vector and a point?
> Why are there two concepts?
>
> Wessel described complex numbers as "vectors" in some space like R^2.
> Argand described complex numbers are "points" in some space like R^2.
>
> What's the difference?
>
> What's wrong with me saying "a vector is a point, and a point is a vector" ?

Vectors can be useful tools for handling geometry, for example in the
complex plane. The key idea is the "position vector" of a point, which
is the vector from the origin to that point. In 3 dimensions, if the
unit vectors along the axes are i, j, k, then the position vector of the
point (x,y,z) is the vector xi + yj + zk. Then, for example, to find
the centroid of a triangle you can just average the position vectors of
its three vertices. Various other geometry can be done in a similar way.

Now here's the confusing bit. It's very common to abbreviate the
vector xi + yj + zk to (x,y,z). That gives us the same notation for
the _coordinates_ of a point and the _components_ of its position
vector. It's not too bad in practice when you get used to it.

Ken Pledger.