3D vector space

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Hi everybody,
I have one question about vectors of R^3:
First of all, a point is described by its co-ordinates (x,y,z).
A vector r is described in this way: r=ax+by+cz ,where {x,y,z} is the standard basis (the numbers a,b,c are the "coordinates" of the vector). But i have seen in several books that vectors can be written like this: r =(x,y,z) where x,y,z are its co-ordinates, as long as the basis is clearly stated. So when we write (x,y,z) we may mean the point, or the vector? For example, if we define a function f:R^3->R^3: (x,y,z)->(f1(x,y,z),f2(x,y,z),f3(x,y,z)) , we can think of f(x,y,z) both as a point and a vector? I mean, can we write
f(a,b,c)=f1(a,b,c)x+f2(a,b,c)y+f3(a,b,c)z ?

I hope this makes sense.
Thanks
 
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Answers and Replies

  • #2
AKG
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Yeah, in general you can treat a point as a vector, and vice versa depending on what you need to do with it. If for some reason it would benefit you to think of f(x,y,z) as a point, then do so, and if you need to treat it as a vector, do so. Normally, there should be no confusion either, since if you write (a,b,c) x (d,e,f) then it's clear that you're treating them as vectors since you're taking their cross product, which is an operation of vectors, not points.
 
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Thanks for your answer.
 
  • #4
HallsofIvy
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What happens is that the "unit" vectors x,y, and z (or, and I think more common i, j, k) are "assumed" in writing (a, b, c). That is, (a, b, c) is just a shorthand way of ax+ by+ cz.

Yes, we do also use (a, b, c) to mean a point so that can be confusing. However, some texts make use of the ambiguity since the vector ax+ by+ cz can be thought of as the vector specifically from the point (0,0,0) to the point (a,b,c). Others, however, prefer the notation <a, b, c> for vectors to distinguish a vector from a point.
 

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