could a vector be also a curved line not only a straight line?
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.
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.
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.
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