Depicting Vectors as Path-Dependent Rays

In summary, the conversation discusses the representation of vectors in physics and math, specifically the problematic depiction of velocity vectors as straight rays. The concept of uniform circular motion and the relationship between velocity and angular velocity is also mentioned. The faulty diagram on a wiki page is used as an example to illustrate the issue. The conversation also touches on the idea of vectors being path dependent and the concept of infinitesimal deltas.
  • #1
vin300
603
4
This is a physics and math problem
The representation of a vector as a straight ray is problematic
Imagine an arc of a circle formed as a result of uniform circular motion.In time Δt, it traces out an arc of length vΔθ such that vΔθ/Δt =vω is the acceleration. v=rω comes from the fact that dx/dt=rdθ/dt and all higher derivatives are derived in exactly the same way with an ω in every next der.
The problem in depicting the velocity vector as a straight ray is clearly depicted in the faulty diagram of this wiki page
http://en.wikipedia.org/wiki/Areal_velocity"
The diagram is clearly wrong for the fact that it does not include the chord within the areal sector and the only solution to this is using the velocity, and in general, all vectors as path dependant (in space or space-time)
 
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  • #2
hell! That's why the deltas are infinitesimal!
Thinking too much is taking a toll
 
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  • #3
rays.

I would agree with the statement that depicting vectors as straight rays can be problematic. While it may be a convenient visual representation, it does not accurately depict the true nature of vectors as path-dependent entities. This is especially evident in cases of circular motion, where the velocity vector changes direction constantly, making it impossible to represent as a straight line.

In physics and mathematics, it is important to accurately represent concepts and phenomena in order to fully understand and analyze them. In the case of vectors, representing them as path-dependent rays allows for a more accurate depiction of their properties and behavior. This is because vectors not only have magnitude and direction, but also a specific path or trajectory that they follow.

Furthermore, the faulty diagram mentioned in the content serves as a clear example of the limitations of depicting vectors as straight rays. It fails to consider the chord within the areal sector, leading to an inaccurate representation of the velocity vector. This highlights the importance of using a more accurate and comprehensive depiction of vectors as path-dependent rays.

In conclusion, while it may be tempting to represent vectors as straight rays for simplicity, it is crucial in physics and math to use more accurate and comprehensive representations, such as depicting vectors as path-dependent rays. This allows for a better understanding and analysis of these fundamental entities in the scientific world.
 

FAQ: Depicting Vectors as Path-Dependent Rays

1. What is the concept of depicting vectors as path-dependent rays?

The concept of depicting vectors as path-dependent rays is an approach used in physics and mathematics to represent vector quantities as arrows or rays with a specific direction and magnitude. It takes into account the path that the vector follows, rather than just its starting and ending points.

2. How is this different from the traditional representation of vectors?

The traditional representation of vectors involves drawing an arrow from the initial point to the terminal point, without considering the path the vector takes. Depicting vectors as path-dependent rays takes into account the path and direction of the vector, providing a more comprehensive understanding of its movement and properties.

3. What are the benefits of using this approach?

Using the path-dependent ray representation of vectors allows for a better visualization and understanding of vector quantities, especially in cases where the path of the vector is important. It also helps in solving problems involving vector operations, as it provides a more accurate representation of the vector's properties.

4. Are there any limitations to this method of depicting vectors?

One limitation of depicting vectors as path-dependent rays is that it can only be used for two-dimensional vectors. It also requires a good understanding of vector operations, as well as the concept of path dependence.

5. How is path dependence related to the representation of vectors?

Path dependence refers to the fact that the path a vector takes can affect its properties, such as its direction and magnitude. Depicting vectors as path-dependent rays takes this into consideration, providing a more accurate representation of the vector's behavior and properties.

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