What the hell are vector components?

[LucasGB]

Are vector components scalars or vectors? Wikipedia.org says they're vectors, Nasa.gov says they're scalars, and I'm just confused.

Arguments saying they're scalars: if you define unit vectors i, j and k, and say components are the numbers multiplying these vectors, then vector components are scalars. Also, the dot product of vectors a and b is defined as axbx+ayby+azbz, where ax,bx,ay... etc. are the components of the vectors, and they are scalars.

Arguments saying they're vectors: often when we work with forces and fields, we decompose a vector into other vectors pointing in convenient directions (such as normal and tangential directions) and say these are their components. In that case, the components are vectors.

So, are vector components scalars or vectors? Thank you for your input!

 

[the_house]

In my opinion, the nasa.gov site is just wrong. Usually a scalar is defined as something that does not change under a rotation (e.g., the norm of a vector). The individual components of a vector, on the other hand, do change under a rotation. The nasa.gov page says that "A quantity which does not depend on direction is called a scalar quantity", and tries to argue that components of a vector do not depend on direction, but to me that argument seems untenable.

Occasionally, people loosely refer to any single number as a "scalar", as opposed to a "vector", which is usually represented as a set of numbers (components, or a magnitude and angles). Thus, it's not surprising that you might hear someone call a component of a vector a "scalar", although I would argue that if you are trying to be precise, it's just not correct. Further, perhaps being even more pedantic, you wouldn't really say that the components of a vector are vectors themselves. They are simply quantities that transform under rotations as components of vectors--not full vectors themselves.

 

[Landau]

It's not really a good question. The components themselves are scalars, but they depend on your choice of basis! First you take a basis (e_1,...,e_n) (assuming finite dimensions for convenience) of the vector space V. Then any vector v\in V can be uniquely expanded v=v_1e_1+...+v_ne_n. Often we just write v=(v_1,...,v_n), but obviously this assumes everyone knows what basis we have in mind (in R^n this is usually the 'standard' basis), for another basis (e'_i) yields other components (v'_i)_i!

This is really the same as the matrix of a linear map: after choosing some basis, a linear map has a unique matrix representation w.r.t. that basis. A different basis yields a different matrix.
Then your question about vector components translates into: what are the matrix entries; scalars or linear maps? Well, the entries themselves are scalars, but they depend on the basis :)

 

[LucasGB]

OK, so the_house thinks they're vectors and Landau thinks they're... I'm sorry Landau, I'm not smart enough to completely understand what you said. "The components themselves are scalars, but they depend on your choice of basis!" So, you're saying they're scalars or vectors?

I understand they're not just any 3 numbers (for 3D vectors), they're 3 numbers which, when multiplied by the basis of choice and added, form the vector in question. But are components defined to be the 3 pure numbers, or the 3 numbers multiplied by their respective basis? This, again, translates into the question: are vector components scalars or vectors?

 

[Landau]

An abstract vector space is just a set (whose elements are called 'vectors'), a field (whose elements are called 'scalars'), and two operations (called 'addition of vectors' and 'scalar multiplication') satisfying some basic rules.

As such, the vector components themselves are scalars, since a vector component is an element of the field.

I think the_house was referring to the physicist's way of differentiating between 'objects that change/stay the same under rotations'. The problem is that in an abstract vector space there need not exist a notion of 'angle', i.e. an inner product, or even a norm, metric, etc.

What answer you find satisfying comes down to what you mean by 'vector', 'scalar' and 'what ARE components'; do you want to use pure mathematical definitions, or some presupposed distinction between vectors and scalars that physicists use (which you will have to spell out). In short: I think I explained in my first post what the components "are", how you want to call them is semantics/depends on your definitions.

 

[suku]

Are vector components scalars or vectors? Wikipedia.org says they're vectors, Nasa.gov says they're scalars, and I'm just confused.

Arguments saying they're scalars: if you define unit vectors i, j and k, and say components are the numbers multiplying these vectors, then vector components are scalars. Also, the dot product of vectors a and b is defined as axbx+ayby+azbz, where ax,bx,ay... etc. are the components of the vectors, and they are scalars.

Arguments saying they're vectors: often when we work with forces and fields, we decompose a vector into other vectors pointing in convenient directions (such as normal and tangential directions) and say these are their components. In that case, the components are vectors.

So, are vector components scalars or vectors? Thank you for your input!

Normal and tangential directions are forms of a vector(if it has anything like that). They are itself vectors and are not components.Components are magnitudes(in x,y,z direction, suppose)that make up a vector.

 

[LucasGB]

As such, the vector components themselves are scalars, since a vector component is an element of the field.

I understand, and this makes a lot of sense to me. But...

Normal and tangential directions are forms of a vector. They are itself vectors and are not components.

If this set of scalars are defined to be the components of the vector, then what should I call this set of vectors which, when added, gives me the vector in question?

 

[Kyouran]

If this set of scalars are defined to be the components of the vector, then what should I call this set of vectors which, when added, gives me the vector in question?

Component vectors

 

[LucasGB]

Component vectors.

OK, you just blew my mind and I feel quite stupid now. But are you sure that's correct, are you sure that's how mathematicians and physicists call it?

 

[the_house]

As Landau says, it is just a matter of semantics since I doubt there's any real confusion about the actual nature of the objects in question.

I was actually unaware of the use of the term "scalar" as described by others in this thread in the context of abstract vector spaces. I just know that physicists often use the word in terms of categorizing how objects transform under some particular transformation or group (e.g., for rotations or Lorentz transformations you can have scalars, vectors, rank 2 tensors, etc.) A scalar, then, is invariant under the given transformation, in which case calling a component of a vector quantity a "scalar" can be confusing since it changes under a rotation. It's useful to know there's another distinct use for the term.

 

[LucasGB]

As Landau says, it is just a matter of semantics since I doubt there's any real confusion about the actual nature of the objects in question.

Yes, but I'm trying to be as rigorous as possible learning these things.

A scalar, then, is invariant under the given transformation, in which case calling a component of a vector quantity a "scalar" can be confusing since it changes under a rotation.

According to Wikipedia,

"In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations (in Newtonian mechanics), or by Lorentz transformations or space-time translations (in relativity)."

So apparently there are two definitions for scalars in physics and mathematics, and this seems to clear up the confusion you described.

God, I love the internet!

 

[the_house]

I'm often amazed at the wealth and quality of information to be found on Wikipedia--even on some pretty advanced physics and math topics.

The internet can be a fantastic resource indeed!

 

[Landau]

So apparently there are two definitions for scalars in physics and mathematics, and this seems to clear up the confusion you described.

God, I love the internet!

Actually, I thought I had mentioned that already

I think the_house was referring to the physicist's way of differentiating between 'objects that change/stay the same under rotations'. The problem is that in an abstract vector space there need not exist a notion of 'angle', i.e. an inner product, or even a norm, metric, etc.
(...)
pure mathematical definitions, or some presupposed distinction between vectors and scalars that physicists use

 

[LucasGB]

Actually, I thought I had mentioned that already

I apologize for not grasping your point before.

 

[Tac-Tics]

I'm often amazed at the wealth and quality of information to be found on Wikipedia--even on some pretty advanced physics and math topics.

The quantity is great. The quality is spotty as hell.

Are vector components scalars or vectors? Wikipedia.org says they're vectors, Nasa.gov says they're scalars, and I'm just confused.

So you have a vector that's written as a*i + b*j + c*k. The scalar components are a, b, and c. The vector components are a*i, b*j, and c*k.

Usually, i, j, and k are your standard basis vectors (the unit vectors pointing along the positive x, y, and z axes). As long as it's understood by context what basis you're using, you can use the scalar components to unambiguously define vectors.

Sometimes, though, it's not worth the hassle of breaking a vector down into a basis and associated scalars. Sometimes you just want a set of vectors which sum to the one you're interested in. Forget bases. That's when you use vector components.

So it's just something you pick up on from context. There is a TON of that going on in every science. The rigorous mathematical details are not important to the problem, and so we end up talking about "vectors in a vector field" when what we really mean is "the evaluation of a vector field at a point" and "infinitesimal neighborhoods of space" when we're really talking about some limit as the volume goes to zero.

 

[Landau]

@Lucas: don't apologize! I probably could have been clearer anyway.

 

[LucasGB]

The quantity is great. The quality is spotty as hell.

So you have a vector that's written as a*i + b*j + c*k. The scalar components are a, b, and c. The vector components are a*i, b*j, and c*k.

Usually, i, j, and k are your standard basis vectors (the unit vectors pointing along the positive x, y, and z axes). As long as it's understood by context what basis you're using, you can use the scalar components to unambiguously define vectors.

Sometimes, though, it's not worth the hassle of breaking a vector down into a basis and associated scalars. Sometimes you just want a set of vectors which sum to the one you're interested in. Forget bases. That's when you use vector components.

So it's just something you pick up on from context. There is a TON of that going on in every science. The rigorous mathematical details are not important to the problem, and so we end up talking about "vectors in a vector field" when what we really mean is "the evaluation of a vector field at a point" and "infinitesimal neighborhoods of space" when we're really talking about some limit as the volume goes to zero.

You could have saved us a lot of trouble if you had posted before.

But now you brought up another point: what's wrong with "infinitesimal neighborhoods of space"? Isn't this simply the notion of differentials?

 

[Tac-Tics]

You could have saved us a lot of trouble if you had posted before.

I got here as fast as I could :)

But now you brought up another point: what's wrong with "infinitesimal neighborhoods of space"? Isn't this simply the notion of differentials?

I have issues with infinitesimals. The reason they are used in physics is partly historical and partly because it's easier to imagine super small bits of dust which act like squares and circles and points at the same time. However, the rigorous basis for them is relatively obscure. I studied up on hyperreal math before, and I'm wasn't convinced it was a good basis for integral calculus.

In my mind, at least, infinitesimals are to standard, limit-based analysis as Newtonian physics is to relativity: for all practical matters, they usually give the same results, but the latter is a more complete model of reality.

 

[LucasGB]

I have issues with infinitesimals. The reason they are used in physics is partly historical and partly because it's easier to imagine super small bits of dust which act like squares and circles and points at the same time. However, the rigorous basis for them is relatively obscure. I studied up on hyperreal math before, and I'm wasn't convinced it was a good basis for integral calculus.

In my mind, at least, infinitesimals are to standard, limit-based analysis as Newtonian physics is to relativity: for all practical matters, they usually give the same results, but the latter is a more complete model of reality.

Very interesting for you to bring this up, because just five minutes ago I was reading this article called "Putting Differentials Back to Calculus", where the authors argue that differentials are very important to calculus and deserve more attention. Although the article is not at all rigorous, it does makes some good points.

http://www.physics.orst.edu/bridge/papers/differentials.pdf

A funny quote from the article: "... many mathematicians think in terms of infinitesimal quantities: apparently, however, real mathematicians would never allow themselves to write down such thinking, at least not in front of the children."

A funny quote from me: "maybe we don't understand differentials well because since they're very small, their behavior is dictated by quantum mechanics."

 

[Kyouran]

It all depends on what you define by "vector components". Some people read the term "vector components" as "(scalar) components of a vector", while others read the term "vector components" as "vector components (of a vector)".
In the first case the word "vector" implies we're talking about components of a vector quantity, and in the second case that same word "vector" refers to the fact that they're talking about vector quantities that make up the vector.

In the end, it's all a matter of preference.