
Vector or not vector?
If a vector is simply "any quantity having magnitude and
direction", then how can a vector's components NOT transform according to the rule of transformation for the components of a vector when we apply a transformation of the coordinate system? Or alternatively, given three numbers (for the 3D case), how can we say that they are NOT the components of a vector in a given coordinate system, if we admit that they transform according to the rule of transformation for vector components when we change the coordinate system? I'm not sure anymore if a vector is simply "any quantity having magnitude and direction", I mean, in a given xy coordinate system (2D case), I can define a vector, say (3,5)  but it's clear that (3,5) are NOT the components of my vector in another coordinate system. Does that mean that my vector is not a vector? I think it just means that (3,5) are not the components of my vector. But then, how do I specify my vector independently of the coordinate system, for isn't a vector an entity that exists regardless of any coordinate system?
I know my question seems confusing. Please read my following analysis before answering:
Vector or not vector?

vector or not a vector?
The definition you told is fine. However, the notion you have about the components of a vector is not ok completely. Actually the components are generally said and the representation you are having, talks about rectangular components of a vector. As the coordinate system changes the rectangular components of a vector also chnage their direction accordingly keeping the magnitude fixed.
If you are talking about a vector, the magnitude remains changed irrespective of the coordinate system. However, as you change the coordinate system or may be if you are rotating the coordinate system then the direction changes and accordingly the direction of the rectangular components change.
I hope this helps!!!!!!!!!

Thank you for your reply, jagabandhu:) I'm still a little confused.:o

well fine!!!!!!!!
I do not know how much you know about the things.
Well may be you can get some ideas in the website
The Velocity Vector and its Components
other wise you may post your curosities in the website
www.askiitians.com :: Index
I hope this may help you out. In the mean time I would like to look back to my notes to clarify more about it!!!!!