What does it mean that vector is independent of coordinate system

In summary, a vector is the same no matter what coordinate system it is represented in, as long as the length and direction remain unchanged. The components may change, but the true identity of the vector lies in its length and direction, not its coordinates. This concept also applies to other mathematical operations involving vectors, such as the cross-product and dot product.
  • #1
aaaa202
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Hi PF, I have always wondered what was meant when my teachers told me that a vector is the same no matter what coordinate system it is represented in. What is it exactly that is the same? I mean the components change. So the only thing that I can see remains the same is the length of the vector. Unless you understand vector as something more abstract than I do. Please explain :)
 
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  • #2
Your description of the vector can change if you express it in a different basis, even if the vector does not.

For example, consider the vector space of polynomials:

X^2+X+2 is a vector in it.
In the basis (1,X,X^2,...) it can be expressed as (2,1,1,0,0,...)
In the basis (1,(X-1),(X-1)^2,...) it can be expressed as (0,-1,1,0,0,...)
 
  • #3
Conceptually, it means a little more than just the values of the components and the length.

It also affects how the rotation works when going from one coordinate system to another. It also spills over into other concepts. The cross-product of two vectors works regardless of the coordinate system you're using, the dot product of two vectors, laws of conservation are still applicable regardless of the coordinate system, etc. (sometimes there's an advantage to using one coordinate system over the other and you choose that alternate coordinate system with no penalty).
 
  • #4
The components change, but the length does not. That is invariant number one, scalar product with itself. Scalar products with other vectors are also invariant. Geometrically, that means that the length and direction of the vector are unaffected by changes of the coordinate system. The length and direction are the true "identity" of the vector, not its coordinates.
 
  • #5
Another way to think about it is that wind blows in a particular direction at a particular speed. That is its "velocity vector". What coordinate system you use, how you measure angles, even whether you measure speed in "miles per hour", "km per hour", or "meters per second", won't affect the wind at all! It will still blow in the same direction at the same speed. It velocity vector is the same no matter what coordinate system you use.
 
  • #6
aaaa202 said:
Hi PF, I have always wondered what was meant when my teachers told me that a vector is the same no matter what coordinate system it is represented in. What is it exactly that is the same? I mean the components change.
Get a blank sheet of white paper. Draw two dots somewhere on that sheet, label them A and B. Draw a directed straight line segment from point A to point B. That directed line segment is a vector. You didn't need a coordinate system to draw it.

Now imagine putting a transparency sheet with grid lines atop that white sheet of paper. Thanks to that grid you can now read off a numerical representation of that vector. Rotate the transparency by 45 degrees and you'll get a different set of numbers. It's still the same vector. All that has changed is how you are representing it. The thing that the vector represents, the displacement from point A to point B hasn't changed.How you choose to represent a vector and the thing that the vector represents are two different things.
 

1. What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is represented by an arrow pointing in the direction of the vector, with the length of the arrow representing the magnitude.

2. How is a vector independent of coordinate system?

A vector is independent of coordinate system because it only has a magnitude and direction, which remain the same regardless of the coordinate system used to describe it. This means that the same vector can be represented by different sets of coordinates, but the vector itself does not change.

3. Can a vector have different components in different coordinate systems?

Yes, a vector can have different components in different coordinate systems. This is because the components of a vector are relative to the coordinate system being used. However, the magnitude and direction of the vector will remain the same.

4. Why is it important for a vector to be independent of coordinate system?

It is important for a vector to be independent of coordinate system because it allows for a consistent and universal way to describe and manipulate vectors. This is especially useful in fields such as physics and engineering, where vectors are commonly used to represent physical quantities.

5. How can we determine if a vector is dependent on coordinate system?

If a vector changes in magnitude or direction when the coordinate system is changed, then it is dependent on coordinate system. However, if the vector remains the same regardless of the coordinate system used, it is independent of coordinate system.

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