Defining Vectors: Scalars vs 1-Dimensional Vectors

[Mr Davis 97]

A very informal definition of a vector could be that it is a tuple of ordered numbers. As such we use the symbol $\mathbb{R}^n$ to refer to the set of vectors with real numbers and n many components. $\mathbb{R}^2$ refers to an element such as $\vec{a} = (x, y)$. And $\mathbb{R}^3$ refers to elements such as $\vec{a} = (x, y, z)$. Thus, why wouldn't it be reasonable to conclude that $\mathbb{R}^1$ or $\mathbb{R}$ would refer to $\vec{a} = x$? Specifically, what makes scalars distinct from one-dimensional vectors? Since they are represented by the same set, why or how are they distinct?

 

[Mark44]

A very informal definition of a vector could be that it is a tuple of ordered numbers. As such we use the symbol $\mathbb{R}^n$ to refer to the set of vectors with real numbers and n many components. $\mathbb{R}^2$ refers to an element such as $\vec{a} = (x, y)$. And $\mathbb{R}^3$ refers to elements such as $\vec{a} = (x, y, z)$. Thus, why wouldn't it be reasonable to conclude that $\mathbb{R}^1$ or $\mathbb{R}$ would refer to $\vec{a} = x$? Specifically, what makes scalars distinct from one-dimensional vectors? Since they are represented by the same set, why or how are they distinct?

You had pretty much the same question in this thread. Wasn't your question answered there?

 

[Mr Davis 97]

You had pretty much the same question in this thread. Wasn't your question answered there?

Well no one answered my last post in that thread, so I thought I would rephrase the question so that I can get a desired answer the first time. jedishrfu noted that "A scalar value can have a positive, zero or negative value whereas a one dimensional vector will always have a positive or zero magnitude and a direction." But I don't see what magnitude and direction have to do with it. Essentially, my question is if a two-dimensional vector is a two-tuple, wouldn't a one-dimensional vector be a one-tuple, i.e., just a single scalar number? I can't find a formal definition of a vector that says it needs a positive magnitude and direction to be a vector.

 

[Mark44]

Instead of defining a vector as an object with direction and magnitude, an alternative definition is to define a vector as something that belongs to a vector space. A vector space is defined by a set of vector space axioms, specifying how vectors can be added or multiplied by scalars.

There's really not much difference between a real scalar and a vector in $\mathbb{R}$. As was mentioned in the other thread, there is a one-to-one mapping between the real numbers and $\mathbb{R}$ (or $\mathbb{R}^1$, if you like).

 

[Mr Davis 97]

Instead of defining a vector as an object with direction and magnitude, an alternative definition is to define a vector as something that belongs to a vector space. A vector space is defined by a set of vector space axioms, specifying how vectors can be added or multiplied by scalars.

There's really not much difference between a real scalar and a vector in $\mathbb{R}$. As was mentioned in the other thread, there is a one-to-one mapping between the real numbers and $\mathbb{R}$ (or $\mathbb{R}^1$, if you like).

So what exactly is the difference between $a = 5$ and $\vec{a} = 5\hat{i}$? I'm assuming they're not interchangeable, so there must be a difference...

 

[tommyxu3]

So what exactly is the difference between $a = 5$ and $\vec{a} = 5\hat{i}$? I'm assuming they're not interchangeable, so there must be a difference...

For $5$ just means an amount, and $5\hat{i}$ means a quantity containing its magnitude and direction. You can say one's speed is $5m/s,$ and you know what it means. But, when you say one's velocity is $5\hat{i}m/s,$ you can get more information in it, that is, where the one is going!

 

[Mark44]

So what exactly is the difference between $a = 5$ and $\vec{a} = 5\hat{i}$? I'm assuming they're not interchangeable, so there must be a difference...

$\vec{a}$ belongs to a vector space, while a doesn't, but otherwise, and as already stated by numerous people in this and the other thread, there's not all that much difference.

For $5$ just means an amount, and $5\hat{i}$ means a quantity containing its magnitude and direction. You can say one's speed is $5m/s,$ and you know what it means. But, when you say one's velocity is $5\hat{i}m/s,$ you can get more information in it, that is, where the one is going!

If you consider that 5 means +5, there's a semblance of direction, as opposed to -5.

 

[tommyxu3]

If you consider that 5 means +5, there's a semblance of direction, as opposed to -5.

I'm not sure if it's correct. When I say the change of the energy is $-5J,$ I won't consider it as $-5\hat{i}$ or any other vector whose magnitude is $5.$

 

[Mark44]

I'm not sure if it's correct. When I say the change of the energy is $-5J,$ I won't consider it as $-5\hat{i}$ or any other vector whose magnitude is $5.$

Where is +5 on the real number line? Where is -5 on the same line? Aren't they in the opposite direction from each other?
The vectors in $\mathbb{R}^1$ map one-to-one with the numbers on the real line, so there's not really much difference between one-dimensional vectors and plain old real numbers.

One problem with defining vectors as things with a magnitude and a direction, is saying which direction $\vec{0}$ points.

 

[pwsnafu]

For $5$ just means an amount, and $5\hat{i}$ means a quantity containing its magnitude and direction.

How are you defining "direction"? Why does the scalar field $\mathbb{R}$ not satisfy your definition but the vector space $\mathbb{R}^1$ satisfy it? What about complex numbers versus complex vector. We are not interested in a physicist's interpretation, rather the mathematics.

 

[aikismos]

A very informal definition of a vector could be that it is a tuple of ordered numbers. As such we use the symbol $\mathbb{R}^n$ to refer to the set of vectors with real numbers and n many components. $\mathbb{R}^2$ refers to an element such as $\vec{a} = (x, y)$. And $\mathbb{R}^3$ refers to elements such as $\vec{a} = (x, y, z)$. Thus, why wouldn't it be reasonable to conclude that $\mathbb{R}^1$ or $\mathbb{R}$ would refer to $\vec{a} = x$? Specifically, what makes scalars distinct from one-dimensional vectors? Since they are represented by the same set, why or how are they distinct?

Scalars ARE distinct from one-dimensional vectors, because not all scalars have sign and therefore direction, and this is largely a question of context. For instance, in dimensional analysis, you might speak of pressure being a unit with the dimensions $m^{1}l^{-1}t^{-1}$ where it's not possible for m, l, or t to have negative values. From a mathematical perspective, scalars aren't always mapped to the reals, but may be mapped the non-negative reals or even non-negative integers. IF AND ONLY IF one maps a scalar value to the set of reals, THEN scalars can be seen as isomorphic the the reals in one dimension.

 

[aikismos]

I'm not sure if it's correct. When I say the change of the energy is $-5J,$ I won't consider it as $-5\hat{i}$ or any other vector whose magnitude is $5.$

Good point! Direction doesn't refer to some abstract concept on the number line in physics, but specifically with directions in one of the three spatial dimensions.

 

[Mark44]

Scalars ARE distinct from one-dimensional vectors, because not all scalars have sign

Sure they do, unless you're talking about complex scalars. The number 5 is positive -- we just don't routinely write the sign.

and therefore direction, and this is largely a question of context. For instance, in dimensional analysis, you might speak of pressure being a unit with the dimensions $m^{1}l^{-1}t^{-1}$ where it's not possible for m, l, or t to have negative values.

But the context of this question was purely mathematical -- scalars vs. one-dim. vectors.

From a mathematical perspective, scalars aren't always mapped to the reals, but may be mapped the non-negative reals

A subset of the reals...

or even non-negative integers.

Also a subset of the reals..

IF AND ONLY IF one maps a scalar value to the set of reals, THEN scalars can be seen as isomorphic the the reals in one dimension.

Whatever real scalars you want to talk about can be mapped one-to-one with some subset of the reals.

 

[Mark44]

Good point! Direction doesn't refer to some abstract concept on the number line in physics, but specifically with directions in one of the three spatial dimensions.

And these spatial dimensions typically are the x, y, and z axes, each of which is a real number line, with positive coordinates off one way, and negatibve coordinates off the opposite direction.

 

[Mark44]

For instance, in dimensional analysis, you might speak of pressure being a unit with the dimensions m 1 l −1 t −1 m^{1}l^{-1}t^{-1} where it's not possible for m, l, or t to have negative values.

But pressure is force per unit area, and force is a vector quantity, with direction.

If I stand on a scale, I am exerting a downward force whose magnitude is mg. From this I can calculate the pressure on the surface of the scale. The scale is exerting a force against me, upward, so we could talk about the pressure of the scale on my feet. I'll grant you that people don't usually talk about pressure in these terms, but the two forces here are oppositely directed, so one of the pressures would be positive and the other negative, it seems to me.

 

[aikismos]

There's really not much difference between a real scalar and a vector in R\mathbb{R}.

Ahh... sorry. I didn't see it in the OP's question that he wanted to confine the question to pure mathematics... thanks!

You keep responding to the OP's question (apparently asked in a previous thread), but not really answering it. You acknowledge they are distinct, but you fail to provide the distinction in a meaningful way. Let me do that for you...

@Mr Davis 97,

"Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field."
(See https://en.wikipedia.org/wiki/Vector_space)

In "pure mathematics", scalars are (and I repeat) NOT the same thing as the reals and can differ as much as real and complex analysis differ. (Complex analysis is an example of sophisticated sort of math that use a more complex building blocks than just the reals, in this case a sum of a real and an imaginary number.) That's because scalars can be ANY field, of which the reals are but the simpliest example. Fields are studied in abstract algebra (See https://en.wikipedia.org/wiki/Abstract_algebra). Fields in essence are mathematical entities that essentially follow the operations you are familiar with when it comes to the reals. Your question can't really be answered until you understand what a field is and see some other examples BESIDES reals. But scalars should never be confused as being essentially the same as reals.

 

[Mark44]

Ahh... sorry. I didn't see it in the OP's question that he wanted to confine the question to pure mathematics... thanks!

You keep responding to the OP's question (apparently asked in a previous thread), but not really answering it.

I and others have responded in both threads that there is no appreciable difference between real scalars and vectors in R¹. The OP asked specifically about real scalars.

You acknowledge they are distinct, but you fail to provide the distinction in a meaningful way. Let me do that for you...

@Mr Davis 97,

"Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field."
(See https://en.wikipedia.org/wiki/Vector_space)

I explicitly excluded complex scalars when I talked about scalars being signed. Of course, in a vector space over some field F, the field can be any scalars, real, complex, members of the set {0, 1}, whatever. In both threads, the scalar field was the real numbers, and my comments were in relation to that.

In "pure mathematics", scalars are (and I repeat) NOT the same thing as the reals and can differ as much as real and complex analysis differ. (Complex analysis is an example of sophisticated sort of math that use a more complex building blocks than just the reals, in this case a sum of a real and an imaginary number.) That's because scalars can be ANY field, of which the reals are but the simpliest example. Fields are studied in abstract algebra (See https://en.wikipedia.org/wiki/Abstract_algebra). Fields in essence are mathematical entities that essentially follow the operations you are familiar with when it comes to the reals. Your question can't really be answered until you understand what a field is and see some other examples BESIDES reals. But scalars should never be confused as being essentially the same as reals.

 

[aikismos]

https://en.wikipedia.org/wiki/Alphabet_(formal_languages)

But pressure is force per unit area, and force is a vector quantity, with direction.

If I stand on a scale, I am exerting a downward force whose magnitude is mg. From this I can calculate the pressure on the surface of the scale. The scale is exerting a force against me, upward, so we could talk about the pressure of the scale on my feet. I'll grant you that people don't usually talk about pressure in these terms, but the two forces here are oppositely directed, so one of the pressures would be positive and the other negative, it seems to me.

You're saying, "pressure in a direction is like speed in a direction." It's not a vector simply by definition. To wit: "Pressure is a scalar quantity..." and "It is incorrect (although rather usual) to say 'the pressure is directed in such or such direction'. The pressure, as a scalar, has no direction." (See https://en.wikipedia.org/wiki/Pressure). I understand your intuitive attempt to make it a vector in the same direction as the force on the surface, however, it's kind of like trying to say that temperature has direction between the particles which form the basis of the statistical measure have direction. Kind of an important conceptual distinction.

In relation to the reals as a foundation for a vector space, I can see why it's appealing to try to equate them and blur over the distinction, but doing so avoids the general purpose of drawing a distinction and that is scalars are the alphabet whereas vectors are the units for the basis (See https://en.wikipedia.org/wiki/Alphabet_(formal_languages)), and vectors are constructed from them. Are letters the same as words? Do we say that digits are the same as numbers? Nope. Digits are used to build a number system. That's the important point to draw from the difference in terminology, even if we just consider the reals, they are used to create a system of infinitely extensible vectors, and are of a different type. I think intuitively the OP wanted to know WHY everyone keeps saying "they're ALMOST the same thing" and no one drew the distinction. One of the goals of undergraduate math is to get the student acclimated to the ability to work with formal systems (See https://en.wikipedia.org/wiki/Formal_system) of all the theories of math, be they number theory, Galois theory, etc.

EDIT: Sorry, I churned that out quickly, and I probably should have restated a few points:

1) If you EXPLICITLY respond to a question three times, and the person asks a fourth time, then it should cast doubt whether you actually answered the question, shouldn't it? :D

2) Scalars are not ESSENTIALLY the same thing as as 1-D vectors, any more than the digits 0-9 are the same as quantities 0-9 even though they are written almost the same, not because they both can't be used for the same process of calculation, (sure they can) but because the ideas behind them are very different, and separate an understanding of numbers from a mere manipulation of them. Calculators can add 1 + 1, but humans can reason about the NATURE of 1 + 1. Failure to differentiate conceptually leads to understandable but invalid conclusions. (See pressure being a scalar above.)

3) Would you tell a student axioms are the same as theorems? I mean, they're essentially both just math statements right? I think to assert that a = a and a = 3 are both equations (sure, they are essentially the same statement, right?) would completely undermine the development of the reasoning process because there's a huge conceptual gulf between a = a (axiom), a = 3 (theorem if started with a different equation), and a := 3 (definition). The same goes for understanding the appropriate relationship between scalar and vector (in any field).

 

[Mark44]

https://en.wikipedia.org/wiki/Alphabet_(formal_languages)

You're saying, "pressure in a direction is like speed in a direction." It's not a vector simply by definition. To wit: "Pressure is a scalar quantity..." and "It is incorrect (although rather usual) to say 'the pressure is directed in such or such direction'. The pressure, as a scalar, has no direction." (See https://en.wikipedia.org/wiki/Pressure). I understand your intuitive attempt to make it a vector in the same direction as the force on the surface, however, it's kind of like trying to say that temperature has direction between the particles which form the basis of the statistical measure have direction. Kind of an important conceptual distinction.

Point taken.

In relation to the reals as a foundation for a vector space, I can see why it's appealing to try to equate them and blur over the distinction, but doing so avoids the general purpose of drawing a distinction and that is scalars are the alphabet whereas vectors are the units for the basis (See https://en.wikipedia.org/wiki/Alphabet_(formal_languages)), and vectors are constructed from them. Are letters the same as words? Do we say that digits are the same as numbers? Nope. Digits are used to build a number system. That's the important point to draw from the difference in terminology, even if we just consider the reals, they are used to create a system of infinitely extensible vectors, and are of a different type. I think intuitively the OP wanted to know WHY everyone keeps saying "they're ALMOST the same thing" and no one drew the distinction. One of the goals of undergraduate math is to get the student acclimated to the ability to work with formal systems (See https://en.wikipedia.org/wiki/Formal_system) of all the theories of math, be they number theory, Galois theory, etc.

I understand the distinction between a vector space and the field associated with that vector space. However, the reals can be considered a vector space over the reals (with the real numbers as also the field), which makes things a bit murkier than it would otherwise be. In regard to the question at hand, I don't consider there to be a dime's worth of difference between a one-dimensional vector in $\mathbb{R}^1$ and a real scalar.

EDIT: Sorry, I churned that out quickly, and I probably should have restated a few points:

1) If you EXPLICITLY respond to a question three times, and the person asks a fourth time, then it should cast doubt whether you actually answered the question, shouldn't it? :D

It depends on the situtation as to whether the question was answered. Sometimes people will ask the same question a number of times, not because they don't understand the answer, but because they want to hear a different answer, one that agrees with their preconceived notions.

2) Scalars are not ESSENTIALLY the same thing as as 1-D vectors, any more than the digits 0-9 are the same as quantities 0-9 even though they are written almost the same, not because they both can't be used for the same process of calculation, (sure they can) but because the ideas behind them are very different, and separate an understanding of numbers from a mere manipulation of them. Calculators can add 1 + 1, but humans can reason about the NATURE of 1 + 1. Failure to differentiate conceptually leads to understandable but invalid conclusions. (See pressure being a scalar above.)

I understand the difference between numbers and our numeric representation of them, but the rest of what you said seems like hand-waving. I maintain that for all practical purposes, real scalars (to include both positive and negative values) are effectively the same as one-dim. vectors. There is a trivial isomorphism between the two sets, so in my view, making a distinction is pedantic.

3) Would you tell a student axioms are the same as theorems? I mean, they're essentially both just math statements right? I think to assert that a = a and a = 3 are both equations (sure, they are essentially the same statement, right?)

No, I wouldn't say that axioms are the same as theorems, for obvious reasons. I also would not say that a = a and a = 3 are "essentially the same statement" for the reason that the first is an identity -- true for any value of a, while the second is true only under a certain condition.

would completely undermine the development of the reasoning process because there's a huge conceptual gulf between a = a (axiom), a = 3 (theorem if started with a different equation), and a := 3 (definition). The same goes for understanding the appropriate relationship between scalar and vector (in any field).