I Distinction between coordinates and vectors

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I am a little confused about the difference between between coordinates and vectors. For example, when first studying vector calculus, you learn about vector fields, which formally are maps ##f: \mathbb{R}^n \to \mathbb{R}^n##, and we say that the function associates to every point in space a vector. However, we clearly see that the domain and codomain of the function are the same, so wouldn't that indicate that points and vectors are not distinct? Is this sloppy notation or is there a real reason why we tend to associate both vectors and points, two seemingly different geometric objects, to the set ##\mathbb{R}^n##?
 

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Vectors are quite different from a point location in a coordinate system although they may be represented by the same (x,y) notation. A vector has magnitude and direction. They can be moved, added together, rotated, magnified, reversed, etc. without changing any locations in a coordinate system. The example vector that starts at the origin and goes to a particular point (x,y) is usually identified with notation like (x,y). But that can cause confusion when axis and coordinates are changed. The vector does not change, but it's representation in (x,y) form will. It might even change to polar coordinates like (r,θ) representing the vector r⋅e. A vector (0.5, 0.7) may go from the point (1,2) to the point (1.5,2.7).
 
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Vectors are quite different from a point location in a coordinate system although they may be represented by the same (x,y) notation. A vector has magnitude and direction. They can be moved, added together, rotated, magnified, reversed, etc. without changing any locations in a coordinate system. The example vector that starts at the origin and goes to a particular point (x,y) is usually identified with notation like (x,y). But that can cause confusion when axis and coordinates are changed. The vector does not change, but it's representation in (x,y) form will. It might even change to polar coordinates like (r,θ) representing the vector r⋅e. A vector (0.5, 0.7) may go from the point (1,2) to the point (1.5,2.7).
Okay, that makes sense. Why are they both represented by ##\mathbb{R}^n## though? Doesn't that create ambiguity?
 

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Okay, that makes sense. Why are they both represented by ##\mathbb{R}^n## though? Doesn't that create ambiguity?
Initially it may seem ambiguous. ##\mathbb{R}^n## has multiple uses -- locations in n-space; vector of magnitude and direction same as from the origin to a point. You will get used to interpreting it in the proper context.

A vector field clearly shows those two different uses of ##\mathbb{R}^n##. Below is one from ##\mathbb{R}^2## to ##\mathbb{R}^2##. Each location has a little vector attached to it. The locations shown cover all the points in [-2,2]x[-2,2]. The vectors shown are all small, within [-0.5, 0.5]x[-0.5,0,5], but they point in all directions.
vectorField.png

This figure is illustrating a mapping from locations in [-2,2]x[-2,2] to vectors in [-0.5, 0.5]x[-0.5,0,5].
 
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Math_QED

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Okay, that makes sense. Why are they both represented by ##\mathbb{R}^n## though? Doesn't that create ambiguity?
Yes that creates ambiguity. In fact, the notation ##\mathbb{R^n}## just represents the set of ##n##-tuples. If we would write ##(\mathbb{R^n}, \mathbb{R},+,.)##, it would be clearer that we mean the vector space over the underlying field of the real numbers with usual vector addition and scalar multiplication. But mathematicians are lazy people (well, not all of them), so most just write ##\mathbb{R^n}## and it should be clear from the context what is meant.
 
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For example, when first studying vector calculus, you learn about vector fields, which formally are maps f:Rn→Rnf: \mathbb{R}^n \to \mathbb{R}^n,
that is not true
Definition. We shall say that a vector field ##v## is defined in a domain ##D\subset\mathbb{R}^m## iff in each local coordinate frame ##x=(x^1,\ldots, x^m)## in ##D## there defined a set of functions ##(v^1,\ldots,v^m)(x)## and under a change of coordinates ##x\mapsto x'=x'(x)## these sets of functions satisfy the equation
$$v^i(x)\frac{\partial x^{i'}}{\partial x^i}=v^{i'}(x')$$
the summation is assumed over repeated indexes (in the left side) and ##v^{i'}## means ##v'^i##; ##1'=1,\quad 2'=2## etc. This is called tensor formalism
 
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StoneTemplePython

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if you have some point ##a##, that is an n-tuple and point ##b## that is also an n-tuple, is there a per se reason that the following are true??

##a + b## is well defined
## 3a ## is well defined, and so on.

Vector spaces exhibit linearity. I don't really think n-tuples do.

A nice little niche inside vector spaces is an inner product space -- and it is here that you get interesting things like notions of length and direction that apply to vectors.

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The distinction between points and vectors can get extra confusing in applications like Machine Learning. There, we are given lots of real valued data points for each feature and we choose to act like they exist in a vector space (or an affine translation of one), with a well defined inner product (typically the standard dot product, though sometimes it comes in a different flavor).
 
Is this sloppy notation or is there a real reason why we tend to associate both vectors and points, two seemingly different geometric objects, to the set ##\mathbb{R}^n##?
It means a position or a vector is expressed as a set of n real numbers. For n=3, position is x=(x,y,z) the vector field may be p=(px,py,pz).
 

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