Vector Components and Translation

[physicsphreak2]

Inspired by a question in Griffiths' E&M book (1.10), I am wondering why the components of a vector do not change when the coordinate system is translated by a constant vector.

I understand that, for instance, the velocity of something moving in a coordinate system won't change if we then transform to another coordinate frame translated by a constant distance. By certainly the position vector would (e.g., if in the first frame an event is at the origin, the components will not be (0,0,0) in any translated frame).

In trying to work out the answer to my question on my own, I'm guessing it has something to do with the fact that translations don't change the BASIS vectors we use to describe the coordinate, whereas rotations or stretches do change the basis vectors? But the example of position as compared to any other vector quantity still confuses me when I think of what would change due to a translation.

Thanks!

 

[WannabeNewton]

Every point gets shifted in the same way under a constant translation so vectors joining two different points will remain unchanged because shifting the two points in the exact same way preserves both the direction and magnitude of the vector.

 

[verty]

The word vector is related to words like convection, convey and vehicle. I think of a vector as the action that moves from A to B. If we apply the same action to something at C, it'll move from C to C+B-A. Call this new point D, D - C = B - A.

Or said another way, if B - A is a vector and we translate by a vector C, (B+C) - (A+C) = B - A is the same vector.

 

[mathskier]

So the vector gets "dragged along" with the coordinate translation? I thought we think of the vector as sitting where it is, and when we try to express it in a new set of coordinates we are trying to find how it would be expressed in a different frame (much like a relativistic transformation).

Also, could somebody address my conjecture that the coordinates only change if we need to re-express the basis vectors of the new coordinate system?

 

[Stephen Tashi]

physicsskier,

Why don't you quote exactly what the book says? What it says isn't clear from your description.

Also, could somebody address my conjecture that the coordinates only change if we need to re-express the basis vectors of the new coordinate system?

The coordinates of a thing are sometimes assigned without any reference to any of its component things and they are sometimes assigned with respect to the component things that make it up. In either case, if thing is a physical entity, changing coordinates does not change the thing and it does not change the component things that make it up. That's why it's called a "change of coordinates", not a change of the thing. Rotating a coordinate system does not change a vector. It only changes the coordinates of the vector. If you think of a physical situation where "a" vector is rotating then it's not really a single vector.

The word vector is related to words like convection, convey and vehicle. I think of a vector as the action that moves from A to B.

This is not a good way to think about vectors in many situations. For example if X(t) is a position vector of the location of an object at time t then what happens between X(1) and X(2) isn't that the object is moved from the origin to X(1) at t=1 and then goes back to the origin and gets moved from the origin back to X(2) at t = 2.

 

[WannabeNewton]

So the vector gets "dragged along" with the coordinate translation? I thought we think of the vector as sitting where it is, and when we try to express it in a new set of coordinates we are trying to find how it would be expressed in a different frame (much like a relativistic transformation).

You can think of it in both ways, in practice it won't matter. These are the passive and active views of transformations. The active view is geometric and can help you see what's going on.

Also, could somebody address my conjecture that the coordinates only change if we need to re-express the basis vectors of the new coordinate system?

Coordinate transformations have nothing to do with basis vectors; they are a mapping of the old coordinate functions to new coordinate functions in terms of the old ones. Basis vectors can transform under a change of coordinates but that is a subsequent result.

Anyways, here's the general transformation rule for vectors under a change of coordinates: $V^{i'} = \frac{\partial x^{i' }}{\partial x^{i}}V^{i}$ (implied summation over $i$). If $x^{i'} = x^{i} + a^{i}$ for constant $a^{i}$ then clearly $V^{i'} = V^{i}$.

 

[physicsphreak2]

physicsskier,

The coordinates of a thing are sometimes assigned without any reference to any of its component things and they are sometimes assigned with respect to the component things that make it up. In either case, if thing is a physical entity, changing coordinates does not change the thing and it does not change the component things that make it up. That's why it's called a "change of coordinates", not a change of the thing. Rotating a coordinate system does not change a vector. It only changes the coordinates of the vector.

Right, this is my point: why do the coordinates NOT change due to translation whereas they DO change due to a rotation or an inversion?

For instance, if I have a vector with components (1,1,1) in some coordinate system, and I then translate my coordinate system by one unit in the x-direction, the components of the vector are STILL (1,1,1)? I am ok with the idea that the vector doesn't change, but shouldn't the coordinates describing it change? But, if I do accept this fact, then wouldn't it imply that no matter how I change my coordinate system the components should be the same because the vector always remains the same?

In which cases do the components change and in which case don't they?

 

[mathskier]

You can think of it in both ways, in practice it won't matter. These are the passive and active views of transformations. The active view is geometric and can help you see what's going on.


Coordinate transformations have nothing to do with basis vectors; they are a mapping of the old coordinate functions to new coordinate functions in terms of the old ones. Basis vectors can transform under a change of coordinates but that is a subsequent result.

Anyways, here's the general transformation rule for vectors under a change of coordinates: $V^{i'} = \frac{\partial x^{i' }}{\partial x^{i}}V^{i}$ (implied summation over $i$). If $x^{i'} = x^{i} + a^{i}$ for constant $a^{i}$ then clearly $V^{i'} = V^{i}$.

Actually, I believed that I am right in this case. We can define the vector components of a vector A as Ai=(A, ei) where the parentheses indicate an inner product and ei are the basis vectors. Translating the coordinates leaves us with the same ei, but any transformation that can be represented by a matrix (rotation, inversion, etc.) changes the components because the basis vectors chance under a change of basis.

 

[WannabeNewton]

They change the components sure but you said "coordinates only change if the basis vectors change". I can change coordinates at will; this has nothing to do with basis vectors. Coordinates are just labels of the points in space. The components of vectors change but those are not coordinates; they are the components of the vectors relative to the basis vectors as represented in those coordinates. I think you are using the word "coordinates" when you really mean "components"; try not to mix the two terms because it will be confusing especially when talking about something like this.

 

[Stephen Tashi]

Right, this is my point: why do the coordinates NOT change due to translation whereas they DO change due to a rotation or an inversion?

You have to clarify whether you are talking about the coordinates of vector as relates to a system of basis vectors or whether you mean some other kind of coordinates. You also aren't making a distinction between "free" and "bound" vectors.

If you are talking about the coordinates of a vector with respect to a set of basis vectors, what would you mean by a "translation" of the coordinates? Do you mean creating a new basis by adding a single constant vector to each of the basis vectors and expressing the vector's coordinates in that new coordinate system? If you mean that, then, in general, the coordinates of the vector would change. Do you mean adding a constant number to each of the coordinates of the vector? That would change the coordinates and also change what vector was being represented.

You should quote what the books says. Your statement of the question isn't clear.

The phrase "the coordinates of a vector" is ambiguous. What coordinate system are you talking about? One guess is that you are talking about a "free" vector representing a directed line segment from a point P to a point Q and that your are defining the "coordinates" of this vector to be the lengths of the projections of line segment PQ on the cartesian coordinate axes. This is not actually a valid coordinate system for individual free vectors because it assigns many different free vectors to the same coordinates. (A line segment RS that is parallel to PQ and of the same length gets the same coordinates.) It is permissible for a coordinate system to assign many different coordinates to the same thing (e.g. polar coordinates) but it should not assign different things the same coordinates.

If you are talking about the above scenario, then, as other posters have said, a linear transformation of the cartesian coordinate representation of the points P,Q does not change the lengths of the projections of PQ on the new coordinate axes. Hence its doesn't change the "coordinates" of the free vector PQ in this pseudo-coordinate system.

 

[verty]

This is not a good way to think about vectors in many situations. For example if X(t) is a position vector of the location of an object at time t then what happens between X(1) and X(2) isn't that the object is moved from the origin to X(1) at t=1 and then goes back to the origin and gets moved from the origin back to X(2) at t = 2.

My one concern is this. If X(1) and X(2) are positions, what is X(2) - X(1), is it a position? Not really, it is a change in position, a displacement, and you are saying (I think) that X(1) is not a change in position because the object need not have been displaced from the origin to X(1). So we have positions and changes in position, positions and displacements. But for me this language is problematic. It would be nice to treat all vectors the same.

So I interpret a vector as the action that (when applied) moves from A to B. I realize there is a language issue here, what is an action when it is not applied? The best I can explain this is, if someone reacted a certain way, one could say "this not the reaction I expected". Now if someone acts a certain way, we could conceivably say "this is not the action I expected". If this is not convincing, I apologize.

X(1) for me is the action that, if applied, would move something from A to B. This allows me to think of all vectors in the same way. If this isn't something you have hit upon, then it won't matter for you.

Where one hits upon this is in deciding what notation one should use for a vector. If (0,1) is a position, perhaps the vector should be written <0,1>. Then we can distinguish vectors and points. But when one works with vector equations, it becomes annoying to have these two strict notations. It is much preferable to think of any point as a vector, with the notation (0,1) meaning the vector. You can see the evidence of this in the description "the position vector".

I just want to say one more thing, a language note. I can phrase "the action that, if applied, would move something from A to B" as "the action that moves from A to B", meaning something like "the wind that blows from the north". We don't say that every day has its own north wind. Every day has the same wind and every action is the same action.

Thank you.

 

[physicsphreak2]

You have to clarify whether you are talking about the coordinates of vector as relates to a system of basis vectors or whether you mean some other kind of coordinates. You also aren't making a distinction between "free" and "bound" vectors.

If you are talking about the coordinates of a vector with respect to a set of basis vectors, what would you mean by a "translation" of the coordinates? Do you mean creating a new basis by adding a single constant vector to each of the basis vectors and expressing the vector's coordinates in that new coordinate system? If you mean that, then, in general, the coordinates of the vector would change. Do you mean adding a constant number to each of the coordinates of the vector? That would change the coordinates and also change what vector was being represented.

You should quote what the books says. Your statement of the question isn't clear.

The phrase "the coordinates of a vector" is ambiguous. What coordinate system are you talking about? One guess is that you are talking about a "free" vector representing a directed line segment from a point P to a point Q and that your are defining the "coordinates" of this vector to be the lengths of the projections of line segment PQ on the cartesian coordinate axes. This is not actually a valid coordinate system for individual free vectors because it assigns many different free vectors to the same coordinates. (A line segment RS that is parallel to PQ and of the same length gets the same coordinates.) It is permissible for a coordinate system to assign many different coordinates to the same thing (e.g. polar coordinates) but it should not assign different things the same coordinates.

If you are talking about the above scenario, then, as other posters have said, a linear transformation of the cartesian coordinate representation of the points P,Q does not change the lengths of the projections of PQ on the new coordinate axes. Hence its doesn't change the "coordinates" of the free vector PQ in this pseudo-coordinate system.

Yes, I realize now that when I said coordinates referring to a vector I meant components of the vector. The *components* are what change. So as mathskier suggests, we can think of a rule like this: if the transformation between the two coordinate systems (not coordinates, but coordinate systems) requires that we change our basis vectors, then the components change. Translations do not require a change of basis vectors, so the components of vectors are the same in all coordinate systems translated uniformly from one another. But rotations, inversions, etc. all require us to change components because we are using basis vectors which have changed.

Correct?

 

[Stephen Tashi]

Translations do not require a change of basis vectors, so the components of vectors are the same in all coordinate systems translated uniformly from one another. But rotations, inversions, etc. all require us to change components because we are using basis vectors which have changed.

Correct?

Neither translations nor rotations change the components of a vector since the components are themselves are vectors. There are many ways to express the same vector as a sum of component vectors. A given vector only has a definite set of components with respect to a given basis.

Neither translations nor rotations "require" a change of basis vectors, but you are correct that it is usually desirable to change the basis vectors that we are using when we change coordinates so that the new basis vectors point along the axes of the new coordinate system.

 

[physicsphreak2]

I meant coordinate system as in the set of axes defined by the our basis vectors. So, the general rule that I tried to state would be true if we be clear that the new coordinate system is described by a basis of vectors pointing along the axes-- then the components only change if the basis vectors change, because otherwise the inner product remains the same in both systems?

 

[Stephen Tashi]

I meant coordinate system as in the set of axes defined by the our basis vectors. So, the general rule that I tried to state would be true if we be clear that the new coordinate system is described by a basis of vectors pointing along the axes

I can agree with that.

-- then the components only change if the basis vectors change, because otherwise the inner product remains the same in both systems?

I don't understand what you mean by "otherwise the inner product remains the same in both systems". This is another tangle of semantics. Technically, the inner product of two vectors in a vector space is a particular function that maps a pair of vectors to a number. This function doesn't turn into a different function if you choose a different basis for the vector space. To implement the inner product in terms of coordinates, we think of it as a function of the vectors coordinates. When we change coordinates, the function that implements the inner product may or may not change. If you change from 2D cartesian coordinates to polar coordinates you must change the function that implements the inner product. If you rotate cartesian coordinates to other cartesian coordinates, you don't have to change the function that implements the inner product.

 

[physicsphreak2]

... since the components are themselves are vectors...

Wait, excuse me? The vector components can be obtained via an inner product of the vector with some set of basis vectors. They are scalars.

 

[Stephen Tashi]

Wait, excuse me? The vector components can be obtained via an inner product of the vector with some set of basis vectors. They are scalars.

No, the components are vectors. Doesn't this book you are studying give any definitions? How does it define a component of a vector?

 

[Stephen Tashi]

I think I found a view of the book online. Griffiths doesn't define "components" like the mathematical world does. He has decided (page 5) that the components of a vector will be "numbers". What the the mathematical world calls a component is a vector iike $A_x \hat{x}$. What Griffiths calls a component is the number $A_x$. You should find a better book.

 

[HallsofIvy]

You seem to be saying that the "mathematical world" would say that the "components" or the vector $a\vec{i}+ b\vec{j}+ c\vec{k}$ are $a\vec{i}$, $b\vec{j}$, and $c\vec{k}$. I disagree with that. I have seen Physics texts that say that but every mathematics text I have would refer to a, b, and c as the "components".

 

[Stephen Tashi]

OK, then must we distinguish between "vector components" and "scalar components"?

Giriffths (page 10) even calls the scalars like $A_x$ the "vector components".


At any rate, parsing out what Griffiths means in terms of the usual definitions of mathematics won't work. He defines a vector to be "any set of three components that transforms like a displacement when you change coordinates". So his "vector" isn't defined in terms of the properties of a "vector space".

(By the way, I actually like Griffiths' book. It's just that anyone who takes his definitions seriously (instead of inutitively) is in for a shock when they take linear algebra.)

 

[verty]

I think "ordinates" (or coordinates I guess) is a better name for the actual numbers. Actually, forget ordinates, coordinates is fine and is perfectly understandable.

 

[WannabeNewton]

Given a vector $V$ and a basis $\{e_i \}$ it is extremely standard to call the scalars $V^{i}$ defined by $V = V^i e_i$ as the components of the vector relative to that basis. I don't see why any confusion would arise from this; the only place where the word component is used in vector analysis is when referring to the above.

 

[George Jones]

Given a vector $V$ and a basis $\{e_i \}$ it is extremely standard to call the scalars $V^{i}$ defined by $V = V^i e_i$ as the components of the vector relative to that basis.

Yes, even in the math world. I have several mathematics texts that define components in this way, ranging from an introductory linear algebra text, to the yellow-and-white grad text "Introduction to Smooth Manifolds" by Lee.

In the physics world, " a vector is something that transforms like a vector" is a special case of Zee's definition of a general tensor, "a tensor is something that transforms like a tensor".

 

[Stephen Tashi]

Given a vector $V$ and a basis $\{e_i \}$ it is extremely standard to call the scalars $V^{i}$ defined by $V = V^i e_i$ as the components of the vector relative to that basis.

It is standard to speak of components (in that sense) "relative to that basis" . The problem is that Griffiths only says "the components" and doesn't specify a basis. As I mentioned, he defines a vector to be " a set of three components ".

Do "the" components of a vector (as scalars) change when we rotate coordinates? This depends on what we understand "coordinates" to mean and whether "rotating coordinates" is suppose to imply an implicit change of basis.

 

[Stephen Tashi]

In the physics world, " a vector is something that transforms like a vector" is a special case of Zee's definition of a general tensor, "a tensor is something that transforms like a tensor".

By way of contrast, in the book Lie Groups, Lie Algebras, and Some of Their Applications by Robert Gilmore, on page 28 there is the delightful definition:

Definition. A tensor is a vector.

 

[dextercioby]

A tensor is an element of a tensor product of vector spaces which is itself a vector space. Thus a tensor is a vector. Likewise, a spinor is a vector.

Added: Books on physics are not sources to learn mathematics. Vectors are learned in linear algebra and put in a new light in differential geometry. A similar example in physics would be: you may learn about the Dirac equation(s) in a regular QM course, but the essence of it/them appears in the QFT course.