I'm jumping in rather late to this very long thread, but there have been a few things said about vectors that deserve comment, IMO.

pressure as a vector quantity
I disagree with Ratch that pressure is a vector quantity. Consider a point at a certain depth beneath water. We know how to calculate the value of the pressure at this depth, but what direction is associated with this pressure? At a point, the pressure is equal in all directions.

matrix or vector multiplication
Some time back in this thread, Caveman gave an example of a 1 X n matrix multiplying an n X 1 matrix, which of course produces a 1 X 1 matrix. For all intents and purposes, such a matrix is a scalar. In the same vein, a vector with only a single component can reasonably be thought of as a scalar.

Alternatively, if you don't like that reasoning, scalars can be thought of as vectors in R\(^{1}\), using their sign to establish direction.

In any case, I don't see the relevance to this discussion of what a particular operation on matrices (e.g., matrix multiplication) or vectors (e.g., vector inner product AKA dot product) has. The usual operations that apply to objects in a vector space are multiplication of a vector by a scalar in some field, and addition of vectors. A vector space has to have additional structure in order to support inner products.

Mark

 

Just one comment, however. It doesn't need a small change in resistance. Any change will alter the PD if the rest of the circuit remains unaltered.

Yes, thanks for the correction, Mr. Studiot. Forgot to use 'even' a small.

About V=IR,
does it not feel inappropriate to define voltage by its effect rather than its cause?
The one difference that i think of between V= IR and potential energy per unit charge is that the former gives the drop in voltage while the latter gives both drop and rise(perhaps, only usually) (?)

 

caveman,

Many things are difficult to measure, but that doesn't mean that they don't exist.

I agree with that, and the point is?

My only point was that ohm-amps is equivalent to joules/coulomb.

Yes, moving charges have that equivalancy. But when a charge is static, a voltage can be present when amp-ohms have no meaning.

If one is defined, so is the other, contrary to your previous statement.

Defined and physically nonexistent.

But then I wouldn't be as clear to everyone else. The needs of the many...

Why not? The resistance or impedance formula is intuitively assumed. You would not want anyone to think you are talking about the electrical linearity of a substance, would you? Ratch

 

Mark44,

I disagree with Ratch that pressure is a vector quantity.

I conceded that point long ago. Although a direction can obtained from the force produced, the scientific community prefers to obtain this direction from the direction of the surface vector and consider pressure a scalar.

We know how to calculate the value of the pressure at this depth, but what direction is associated with this pressure?

The direction of a zero vector.

At a point, the pressure is equal in all directions.

Correct, the direction is undefined.

Some time back in this thread, Caveman gave an example of a 1 X n matrix multiplying an n X 1 matrix, which of course produces a 1 X 1 matrix. For all intents and purposes, such a matrix is a scalar.

Dimensionally and conceptionally it is not, even though the element may have the same value as the dot product.

In the same vein, a vector with only a single component can reasonably be thought of as a scalar.

Same answer as above. Even though the vector has only one direction of movement, the direction can be pointing anywhere, whereas a scalar does have any direction associated with it.

Alternatively, if you don't like that reasoning, scalars can be thought of as vectors in R, using their sign to establish direction.

Then you are giving a direction to a scalar which it is not supposed to have.

In any case, I don't see the relevance to this discussion of what a particular operation on matrices (e.g., matrix multiplication) or vectors (e.g., vector inner product AKA dot product) has.

We were talking about their equivalency.

The usual operations that apply to objects in a vector space are multiplication of a vector by a scalar in some field, and addition of vectors.

In addition to the cross product AxB, the scalar triple product (box product) A.(BxC), and the vector triple product Ax(BxC) .

A vector space has to have additional structure in order to support inner products.

I do not understand that statement. Ratch

 

I was speaking about charges that do not move, in which case I*R has no meaning.

Hypothesis contrary to fact. There are no "charges that do not move." There is no perfect insulator. Your conclusion is not valid.

http://www.justsomelyrics.com/220889/Huey-Lewis-&-The-News-Perfect-World-Lyrics

 

thingmaker3,

Hypothesis contrary to fact. There are no "charges that do not move."

On the quantum level, no. But we were talking about the macro world of circuits, so your statement is not applicable.

There is no perfect insulator. Your conclusion is not valid.

If the insulator is so good and so thick that the leakage is undetectable, then it becomes a matter of philosophic argument as to whether it is perfect. Therefore my conclusion is valid.

http://www.justsomelyrics.com/220889...t-World-Lyrics

I don't take wisdom from a song smith. Ratch

 

On the quantum level, no. But we were talking about the macro world of circuits, so your statement is not applicable.

My statement is perfectly applicable. No perfect insulator means "no perfect insulator." If there is one, show it to me now.

If the insulator is so good and so thick that the leakage is undetectable, then it becomes a matter of philosophic argument as to whether it is perfect. Therefore my conclusion is valid.

You have already conceded that undetectability does not equate to non-existence. Philosophy is not relevant. Your conclusion is invalid.[/quote]

I don't take wisdom from a song smith.

The song was for fun, not for wisdom. What you "take" is not of interest to me.

 

I did know what you meant. I was just wishing that you would refer to it by its correct name.

He did. Look up the definition of "definition."

 

thingmaker3,

He did. Look up the definition of "definition."

Won't do any good. You have to lead me by the hand. Ratch

 

Since the forum editor apparently doesn't support comments nested more than one level deep, to give some context, I have marked quotes from myself, quotes from Ratch, and new comments from myself.
Quote from Mark:
I disagree with Ratch that pressure is a vector quantity.
Quote from Ratch:
I conceded that point long ago. Although a direction can obtained from the force produced, the scientific community prefers to obtain this direction from the direction of the surface vector and consider pressure a scalar.

Quote from Mark:
We know how to calculate the value of the pressure at this depth, but what direction is associated with this pressure?

Quote from Ratch:
The direction of a zero vector.
Comment from Mark: I thought you said you conceded that point, that pressure was a vector. Evidently not. In any case, the zero vector is useful only to serve as an additive or multiplicative identity in a vector space. In the context of pressure I can't see that it brings any clarity to the discussion.


Quote from Mark:
At a point, the pressure is equal in all directions.
Quote from Ratch:
Correct, the direction is undefined.
Comment from Mark: And would therefore seem to be a meaningless concept, or at least one that is not very useful.


Quote from Mark:
Some time back in this thread, Caveman gave an example of a 1 X n matrix multiplying an n X 1 matrix, which of course produces a 1 X 1 matrix. For all intents and purposes, such a matrix is a scalar.
Quote from Ratch:
Dimensionally and conceptionally it is not, even though the element may have the same value as the dot product.
Comment from Mark: You brought that up earlier in the thread. For the dimensional part, are you saying that a 1 X 1 matrix differs from a scalar simply because it is a matrix? I will agree that a matrix is fundamentally different from a scalar, but as I said earlier, a 1 X 1 matrix is otherwise undistinguishable from a scalar. As far as conceptually, I can conceive that 1 X 1 matrices behave similarly to scalars, so since I can conceive this, they are not conceptually different to me. If they are conceptually different to you, please show me at least one difference (other than one is a matrix and the other is not).


Quote from Mark:
In the same vein, a vector with only a single component can reasonably be thought of as a scalar.
Quote from Ratch:
Same answer as above. Even though the vector has only one direction of movement, the direction can be pointing anywhere, whereas a scalar does have any direction associated with it.
Comment from Mark: As I pointed out earlier, the direction of a scalar can be thought of as the direction from the origin (0) to the point on the number line that represents the scalar (assuming that the scalars are coming from the field of real numbers). I didn't think I had to explain that further.


Quote from Mark:
Alternatively, if you don't like that reasoning, scalars can be thought of as vectors in R, using their sign to establish direction.
Quote from Ratch:
Then you are giving a direction to a scalar which it is not supposed to have.
Comment from Mark: See above.


Quote from Mark:
In any case, I don't see the relevance to this discussion of what a particular operation on matrices (e.g., matrix multiplication) or vectors (e.g., vector inner product AKA dot product) has.
Quote from Ratch:
We were talking about their equivalency.
Comment from Mark: OK

Quote from Mark:
The usual operations that apply to objects in a vector space are multiplication of a vector by a scalar in some field, and addition of vectors.
Quote from Ratch:
In addition to the cross product AxB, the scalar triple product (box product) A.(BxC), and the vector triple product Ax(BxC) .
Comment from Mark: These operations don't apply to vector spaces in general. All three of these products are specific the vector spaces of dimension 3.


Quote from Mark:
A vector space has to have additional structure in order to support inner products.
Quote from Ratch:
I do not understand that statement. Ratch
Comment from Mark: The vector space has to have an inner product defined for it. By definition, a vector space will have an addition operation and an operation by which a scalar (an element of a specific field) can multiple a vector in the vector space. The additional structure I referred to is some definition of an inner product, which can be as simple as the familiar dot product in Rn, or a product involving an integral if the vector space is a space of functions such as cos(x), cos(2x), cos(3x), and so on.

 

Mark44,

Quote from Ratch:
I conceded that point long ago. Although a direction can obtained from the force produced, the scientific community prefers to obtain this direction from the direction of the surface vector and consider pressure a scalar.

Quote from Mark:
We know how to calculate the value of the pressure at this depth, but what direction is associated with this pressure?

Quote from Ratch:
The direction of a zero vector.
Comment from Mark: I thought you said you conceded that point, that pressure was a vector. Evidently not.

I said that I conceded that pressure should be considered a scalar, not a vector. Read what I said again.

In any case, the zero vector is useful only to serve as an additive or multiplicative identity in a vector space. In the context of pressure I can't see that it brings any clarity to the discussion.

It means that if there is no object submerged, there is no surface for the pressure to push against. The zero vector shows that no force is applied to a submerged object and no direction is indicated. In other words, it has a physical meaning.

Quote from Mark:
At a point, the pressure is equal in all directions.
Quote from Ratch:
Correct, the direction is undefined.
Comment from Mark: And would therefore seem to be a meaningless concept, or at least one that is not very useful.

What is there about directionless static pressure that you do not understand?

Quote from Mark:
Some time back in this thread, Caveman gave an example of a 1 X n matrix multiplying an n X 1 matrix, which of course produces a 1 X 1 matrix. For all intents and purposes, such a matrix is a scalar.
Quote from Ratch:
Dimensionally and conceptionally it is not, even though the element may have the same value as the dot product.
Comment from Mark: You brought that up earlier in the thread. For the dimensional part, are you saying that a 1 X 1 matrix differs from a scalar simply because it is a matrix?

Yes, even when the matrix element has the same value as the scalar.

I will agree that a matrix is fundamentally different from a scalar, but as I said earlier, a 1 X 1 matrix is otherwise undistinguishable from a scalar.

You can say it, but I don't agree with it.

As far as conceptually, I can conceive that 1 X 1 matrices behave similarly to scalars, so since I can conceive this, they are not conceptually different to me.

To each his/her own conceptions.

If they are conceptually different to you, please show me at least one difference (other than one is a matrix and the other is not).

Sure, a matrix is a ordered set of elements defined to positions in rows and columns. Just because a matrix reduces down to a single element whose value is the same as a scalar does not mean they are equivalent. The single matrix element still has row/column position associated with it. Just because you chose to ignore this positional information for even a good reason does not mean that a 1x1 matrix is equivalent to a scalar.

Quote from Mark:
In the same vein, a vector with only a single component can reasonably be thought of as a scalar.
Quote from Ratch:
Same answer as above. Even though the vector has only one direction of movement, the direction can be pointing anywhere, whereas a scalar does not have any direction associated with it.
Comment from Mark: As I pointed out earlier, the direction of a scalar can be thought of as the direction from the origin (0) to the point on the number line that represents the scalar (assuming that the scalars are coming from the field of real numbers). I didn't think I had to explain that further.

Apologies for missing a word (colored red above) in my last response.

The number line is not a physical direction. It is conceptual drawing showing the relative magnitude and sign of real numbers. Therefore any reference to a number on the "number line" is a scalar, not a vector.

Quote from Mark:
Alternatively, if you don't like that reasoning, scalars can be thought of as vectors in R, using their sign to establish direction.
Quote from Ratch:
Then you are giving a direction to a scalar which it is not supposed to have.
Comment from Mark: See above.

See answer above.

Quote from Mark:
The usual operations that apply to objects in a vector space are multiplication of a vector by a scalar in some field, and addition of vectors.
Quote from Ratch:
In addition to the cross product AxB, the scalar triple product (box product) A.(BxC), and the vector triple product Ax(BxC) .
Comment from Mark: These operations don't apply to vector spaces in general. All three of these products are specific the vector spaces of dimension 3

And are they not defined for vector spaces of dimension 4 or higher? It appears to me that they are.

Quote from Mark:
A vector space has to have additional structure in order to support inner products.
Quote from Ratch:
I do not understand that statement. Ratch
Comment from Mark: The vector space has to have an inner product defined for it. By definition, a vector space will have an addition operation and an operation by which a scalar (an element of a specific field) can multiple a vector in the vector space. The additional structure I referred to is some definition of an inner product, which can be as simple as the familiar dot product in Rn, or a product involving an integral if the vector space is a space of functions such as cos(x), cos(2x), cos(3x), and so on.

Well, you lost me. I only know about the dot product. Ratch

 

thingmaker3,

Quote:
Hypothesis contrary to fact. There are no "charges that do not move."

Quote:
Originally Posted by Ratch
On the quantum level, no. But we were talking about the macro world of circuits, so your statement is not applicable.

My statement is perfectly applicable. No perfect insulator means "no perfect insulator." If there is one, show it to me now.

You are getting confused. My answer was in response to you saying that all charges move. As to perfection, if I can show you something so good that you cannot detect any imperfections, then I have showed you a perfect something. It is a philosophic statement that does have meaning.

You have already conceded that undetectability does not equate to non-existence.

When? Where?

The song was for fun, not for wisdom. What you "take" is not of interest to me.

Fair enough. Funny is in the eye of the beholder. Ratch

 

I said that I conceded that pressure should be considered a scalar, not a vector.

It means that if there is no object submerged, there is no surface for the pressure to push against. The zero vector shows that no force is applied to a submerged object and no direction is indicated. In other words, it has a physical meaning.

What is there about directionless static pressure that you do not understand?

Your first sentence above seems to contradict what you're saying in the second paragraph: that pressure is a scalar on the one hand, and that it is a vector quantity on the other. How can you say that you conceded something when you continue to assert it?
Regarding directionless static pressure, I understand this concept pretty well. I think we're arguing about a distinction without a difference; namely that there is no direction associated with the pressure (my view) and that the direction is undefined (your view as I see it).

Sure, a matrix is a ordered set of elements defined to positions in rows and columns. Just because a matrix reduces down to a single element whose value is the same as a scalar does not mean they are equivalent. The single matrix element still has row/column position associated with it. Just because you chose to ignore this positional information for even a good reason does not mean that a 1x1 matrix is equivalent to a scalar.

I understand the differences between matrices and scalars. To a mathematician, however, the set of all 1x1 matrices with entries from the reals is homeomorphic to the set of real numbers. IOW, there is a mapping from the set of these matrices to the real numbers that is one-to-one, onto, continuous, and the inverse map is continuous. This is a more precise way of saying that the two sets are, for all intents and purposes, indistinguishable.

If you have an ordered set of two, three, four, or more things, the order in which the elements appear is important. OTOH, order is unimportant if you can have only one thing in your ordered set.

Quote from Mark:
In the same vein, a vector with only a single component can reasonably be thought of as a scalar.
Quote from Ratch:
Same answer as above. Even though the vector has only one direction of movement, the direction can be pointing anywhere, whereas a scalar does not have any direction associated with it.
Comment from Mark: As I pointed out earlier, the direction of a scalar can be thought of as the direction from the origin (0) to the point on the number line that represents the scalar (assuming that the scalars are coming from the field of real numbers). I didn't think I had to explain that further.

Apologies for missing a word (colored red above) in my last response.

A one-d vector can't "be pointing anywhere." There are only two possible directions for such a vector. It's easy to show that the set of vectors in R1 is homeomorphic to the reals, making these two sets structurally equivalent, at least in the minds of mathematicians.

The number line is not a physical direction. It is conceptual drawing showing the relative magnitude and sign of real numbers. Therefore any reference to a number on the "number line" is a scalar, not a vector.

Agreed, the number line is not a direction, physical or otherwise. However, the location of a real number on the number line is determined by two things: its magnitude, and its sign. 3.5 is located 3.5 units to the right of zero; -2 is
located 2 units to the left of zero.

And are they not defined for vector spaces of dimension 4 or higher? It appears to me that they are.

All three of the products you listed (cross product, triple product, box product) are defined only for vectors in R3.

Well, you lost me. I only know about the dot product.

The dot product for vectors in Rn is only one example of an inner product for vector spaces. You can find out more in the wikipedia article on inner product spaces.
Mark

 

There are two distinctly different versions of pressure.

1. If you are applying a force to an object, it is sometimes useful to look at the force per area. For example, pressing with the same force with a nail on a wall has more penetrating capability that if you used a book. Note that sometimes, however, the pressure is defined as a scalar in this sense as well. I would be defined like this: "Pressure is the magnitude of the force normal to the surface divided by the area over which the force is applied". Basically, the directional nature of pressure is removed via definition.

2. The second version is always a scalar. This describes the magnitude of the force that would be applied to a small test surface in a compressed fluid divided by the area of the test surface. The force will always be normal to the test surface. I have also seen this defined by making the area term in the equation a vector with direction normal to the test surface.

If you look at the results predicted, they are all the same. The definitions that define direction in the definitions are typically what would be seen in older physics texts before vector analysis became popular. The newer terms (like with vectored area, for example) are obviously attempts to create consistency across all of the physics equations. They are all right, though.

 

I am no good with vectors, but should a scalar be considered a vector at all?

What would happen to scalar multiplications? If we can consider one scalar as a vector then can we not also consider the real numbers as vectors as well? In this case what happens to say, a multiplication of a vector with a scalar '-1', which should change its direction. Now that this -1 is being treated as a vector we can not go for a scalar multiplication (can we?). So what else do we go for dot product?(will this not make the actual vector a scalar?) OR a vector product?(Which will not reverse the vectors direction direction and we land up with a wrong answer)

This was the only example that I could come up with, but does it not prove that a scalar should never be considered a vector?

 

I have avoided the pressure debate until now, partly because it has nothing to do with the original thread subject (voltage) and partly because the original hydraulic analogy for electric current (back in the late 19 century) flow was to do with static head, not pressure.

I had hoped to go on to encompass this when I developed the example I presented in post 64.

However I received such flippant replies I decided not to bother.

In particular the question I asked at the end, in preparation for the next stage, has relevance to pressure (and many other things).

In many areas of physical analysis the subject is presented at an elementary level as an averaging or aggregating process, so for instance pressure is the average force per unit area and current density is the average per unit area and so on.

The next step is to shrink the test area or section to zero and apply the calculus to the limiting process. By doing this we can derive many useful (differential) equations in mathematical physics or engineering.

For instance by doing this in a fluid we get the 'pressure at a point' and the fluid flow equations. By doing this in a solid we get the stress at a point, which has 9 components because, unlike a fluid, a solid can support shear stress.

These are fascinating subjects to explore but i believe they belong in another thread, not punctuated by flippant remarks.

 

Mark44,

Your first sentence above seems to contradict what you're saying in the second paragraph: that pressure is a scalar on the one hand, and that it is a vector quantity on the other. How can you say that you conceded something when you continue to assert it?
Regarding directionless static pressure, I understand this concept pretty well. I think we're arguing about a distinction without a difference; namely that there is no direction associated with the pressure (my view) and that the direction is undefined (your view as I see it).

I think Caveman in above post sums up my thinking on the subject. As he says, the history of pressure definition has not been consistent. I have agreed to go with the modern thinking about the definition of pressure.

Quote:
Originally Posted by Ratch
Sure, a matrix is a ordered set of elements defined to positions in rows and columns. Just because a matrix reduces down to a single element whose value is the same as a scalar does not mean they are equivalent. The single matrix element still has row/column position associated with it. Just because you chose to ignore this positional information for even a good reason does not mean that a 1x1 matrix is equivalent to a scalar.

I understand the differences between matrices and scalars. To a mathematician, however, the set of all 1x1 matrices with entries from the reals is homeomorphic to the set of real numbers. IOW, there is a mapping from the set of these matrices to the real numbers that is one-to-one, onto, continuous, and the inverse map is continuous. This is a more precise way of saying that the two sets are, for all intents and purposes, indistinguishable.

If you have an ordered set of two, three, four, or more things, the order in which the elements appear is important. OTOH, order is unimportant if you can have only one thing in your ordered set.

Important is in the eye of the beholder who is working on an application. I have never seen a vector dot product defined as a matrix multiplication, although it gives correct results if its matrix designation is ignored. I have seen a cross product defined as a determinant of a matrix because it gives correct dimension results.

A one-d vector can't "be pointing anywhere." There are only two possible directions for such a vector. It's easy to show that the set of vectors in R1 is homeomorphic to the reals, making these two sets structurally equivalent, at least in the minds of mathematicians.

There are only two possible choices of movement once a direction is chosen. In a rectangular system, the single direction can be the x,y, or z axis.

Originally Posted by Ratch
The number line is not a physical direction. It is conceptual drawing showing the relative magnitude and sign of real numbers. Therefore any reference to a number on the "number line" is a scalar, not a vector.

Agreed, the number line is not a direction, physical or otherwise. However, the location of a real number on the number line is determined by two things: its magnitude, and its sign. 3.5 is located 3.5 units to the right of zero; -2is
located 2 units to the left of zero.

Agreed, but if it has no direction then it is not a vector. The fact that you can locate a position on the number line with one parameter does not make it a 1-dimensional vector.

All three of the products you listed (cross product, triple product, box product) are defined only for vectors in R3.

R3 represents the real physical world. But why cannot, say the dot product, be defined for R4 and beyond?

The dot product for vectors in Rn is only one example of an inner product for vector spaces. You can find out more in the wikipedia article on inner product spaces.

OK. Ratch

 

My answer was in response to you saying that all charges move. As to perfection, if I can show you something so good that you cannot detect any imperfections, then I have showed you a perfect something. It is a philosophic statement that does have meaning.

Argumentum ad ignorantiam. My inability to detect imperfection does not indicate lack of imperfection. Your "philosophic statement" is empty rhetoric, a semantics game, and therefore irrelevant. Your claim of a perfect insulator is an extraordinary claim, and the burden of extraordinary proof is squarely upon your own shoulders.

1 Volt = 1,000,0000 YottaOhms * .000001 YoctoAmps

I've got a lot more zeros where those came from. Show me your "perfect" insulator.

 

thingmaker3,

Argumentum ad ignorantiam. My inability to detect imperfection does not indicate lack of imperfection.

For any meaningful purpose, it does.

Your "philosophic statement" is empty rhetoric, a semantics game, and therefore irrelevant.

Not so. It is a practical concept.

Your claim of a perfect insulator is an extraordinary claim, and the burden of extraordinary proof is squarely upon your own shoulders

I accepted that challenge stated my reasoning. If you don't accept it, I am not responsible for your doubts.

1 Volt = 1,000,0000 YottaOhms * .000001 YoctoAmps

I've got a lot more zeros where those came from.

But no way to measure or verify them.

Show me your "perfect" insulator.

As far as perfection can be achieved, I did. Ratch

 

Golly. How could anyone argue with that?

I'm finished with this thread, save to monitor for compliance with our forum rules. Best wishes for everyone else to enjoy.