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I just need an answer in a paragraph . I will be very grateful if i get a quick answer

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True vectors are geometric invariants. They have different representations in different co-ordinate systems but the actual geometric object does not change.

In classical physics, the force vector for the most part is a 'true' vector in the sense that you need 0-coordinates to describe it. I could say that a force of 10N is acting on your centre of mass and I wouldn't need any co-ordinates to describe it. However, do note that force is actually a bound vector.

However, there do exist some vectors which do not behave in such a way. These vectors are called pseudo-vectors. The examples of pseudo vectors are torque and angular momentum.

Furthermore, there are some quantities which behave in an even more strange way, that is they behave as a vector only for infinitesimal changes. An example of such quantity is the angular velocity vector. Edit: Do note that angular velocity vectors are not true vectors. They are an object known as pseudo-vectors.


References:

Angular momentum vectors: The answer by user JEB in this stack post

The more mathematical side of pseudo-vectors: This youtube video

Force being a bound vector: This wiki

About angular velocity is a vector only for small rotations: This reddit post

An introduction to tensors which discussed about the geometric invariance of vectors: Maththebeautiful

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  • $\begingroup$ why introduce complications like pseudo-vectors? And why would you thing that angular velocity is related to infinitesimal change? Is regular velocity a vector only for infinitesimal changes? $\endgroup$ Sep 12, 2020 at 21:24
  • $\begingroup$ If you think that is complicated, it's probably time to gauge how interested you are in this subject. The true formulations of even the basic ideas require complicated mathematics. It does no good to hide the elephant in the room. Other than that, the 'angular velocity' is not a vector for 'finite' rotations.. only when you take the limit as the angle change becoming very small does it show vector like properties. $\endgroup$ Sep 12, 2020 at 21:29
  • $\begingroup$ If someone needs to ask that level of question about vectors, he or she will find pseudo-vector overly complicated... And it is perfectly possible to define finite rotations using angular velocity vectors (take $\vec \omega$ to be constant) just like it is possible to define finite translation using linear velocity vectors. $\endgroup$ Sep 12, 2020 at 21:34
  • $\begingroup$ I don't know about you, but when I first started learning physics I found it extremely confusing that some online resources referred it as a vector whilst my highschool text referred it as a scalar. So, learning about pseudo-vectors was a true englithenment for me. You can't really say that you understand this unless you understand it's true nature, further it really leads into how important math is for physics. I still disagree that you can try finite rotations as angular velocity vector because the angles start obeys the vector space axioms in the limiting case. $\endgroup$ Sep 12, 2020 at 21:37
  • $\begingroup$ I wish this site to become a place where people could broaden their scope and vision of the subject rather being battered on by the same rephrased from paragraphs from textbook. $\endgroup$ Sep 12, 2020 at 21:38
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Yes, the value of vector quantity will change if the reference axis is changed.
Take for example the displacement vector. If you shift your origin, then the value of the displacement vector will change.
Also, you can consider the angular momentum vector, it's very much dependent on the axis you are choosing.
The above two are just basic examples. In general, the statement is true.

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It depends. If your vector is defined relative to a co-ordinate system - if it is an absolute position vector, for example - then it will change if the origin of the co-ordinate system is moved. However, if the axes of the co-ordinate system are rotated but the origin stays in the same place then the position vector itself does not change, although its components will do.

On the other hand, if a vector is defined in a way that is independent of any co-ordinate system - if it is the position of one object relative to another, for example - then the vector itself does not change. Its components (which are just an arbitrary way that we describe a vector) will change if we change the co-ordinate system, but the vector itself does not change.

The distinction between the vector itself and its co-ordinates relative to a particular co-ordinate system is important in physics theories such as general relativity.

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