- #1
MathematicalPhysicist
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could a vector be also a curved line not only a straight line?
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Could a vector be also a curved line not only a straight line?
- Thread starter MathematicalPhysicist
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In summary, a vector cannot be a curved line as it is simply a geometric object representing a magnitude and direction, not a shape. However, it is possible to talk about vectors in curved fields or surfaces, but they would still be represented as straight lines.
- #2
HallsofIvy
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In the sense in which vectors are geometric objects at all, no.
(The general, abstract, definition of a "vector space" says nothing about a vector having a "shape" so the question would not arise.)
In dealing with vectors in terms of coordinates, vectors are always derivatives- since the derivative is a way of "linearizing" a curve, it would still be most sensible to talk of vectors in terms of straight lines.
It is possible to talk about vectors on the sphere or other curved surface- but then you have tangent vectors- the vectors at a given point lie on the tangent plane at that point, not on the surface itself.
(The general, abstract, definition of a "vector space" says nothing about a vector having a "shape" so the question would not arise.)
In dealing with vectors in terms of coordinates, vectors are always derivatives- since the derivative is a way of "linearizing" a curve, it would still be most sensible to talk of vectors in terms of straight lines.
It is possible to talk about vectors on the sphere or other curved surface- but then you have tangent vectors- the vectors at a given point lie on the tangent plane at that point, not on the surface itself.
- #3
NeutronStar
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I agree with what Halls of Ivy said. But as a further comment I would like to add that it is quite possible to talk about a vector field in which a plot of many vectors can indeed describe a curved line.
So while an individual vector cannot be curved, a vector field most certainly can be curved. In fact, this concept is used all the time.
I think the problem with thinking of a single vector as being curved it that is assumes that the graphic arrow representing the vector has positional value. It does not. It merely represent an idea of magnitude. So it wouldn't make any sense to draw a single vector as a curved line.
Even when drawing vector fields it is understood that at any given point the vector has a particular magnitude. Therefore if you draw the vector for any particular point you would represent its magnitude as the length of a straight line. The angle of the vector would also be a specific value associated with the direction of the vector at that particular point.
That's my 2 cents. I hope I understood your question correctly.
- #4
arcnets
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A vector AB just gives the position of B relative to A. It doesn't say by which path you get there. If you like, you can draw the geometric representation of AB straight, curved, wiggled, blue, red, dotted... it doesn't matter.
1. Can a vector be represented by a curved line instead of a straight line?
Yes, a vector can be represented by a curved line in certain cases. Vectors can be defined as quantities that have both magnitude and direction, and they can also change direction over time. In some cases, a curved line may better represent the changing direction of a vector.
2. What is the difference between a straight line vector and a curved line vector?
The main difference between a straight line vector and a curved line vector is the direction of the vector. A straight line vector has a constant direction, while a curved line vector can change direction over time. Additionally, a curved line vector may have a varying magnitude, whereas a straight line vector has a constant magnitude.
3. How are curved line vectors used in physics and engineering?
Curved line vectors are commonly used in physics and engineering to represent the motion of objects. For example, in mechanics, a curved line vector can represent the path of a moving object, taking into account the changes in its direction and speed. In fluid dynamics, curved line vectors are used to represent the flow of fluids.
4. Can a curved line vector be converted into a straight line vector?
Yes, a curved line vector can be approximated by a series of straight line vectors. This is often done in mathematics and computer science, where curved line vectors are represented as a series of shorter straight line vectors, allowing for easier calculations and computations.
5. Are there any limitations to using curved line vectors?
Yes, there are some limitations to using curved line vectors. While they can accurately represent the changing direction of a vector, they may not always be the most efficient or practical way to represent a vector, particularly in mathematical and computational applications. Additionally, curved line vectors may be more difficult to visualize and understand compared to straight line vectors.
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