Vectors along different coordinate axes

[Monsterboy]

Homework Statement

A situation may be described by using different sets of coordinate axes having different orientations. Which of the following do not depend on the orientation of the axes?
A. The value of a scalar
B. Component of vector
C. A vector
D. The magnitude of vector

Relevant Equations

None

The answer in the textbook are options A, C and D.

I understand why it is option A, because it is a scalar, I also get that option D is correct because the magnitude of a vector doesn't depend on the coordinate axes. I don't get how option C could be correct. If option C is correct why not D as well ?

A vector is defined by it's magnitude and direction. The direction is defined by the angle it makes with a coordinate axis. So if the coordinate axis gets changed and the angle between the vector and the new coordinate axis remains the same as before, the vector is said to be unchanged right ? i.e not affected by the change in the coordinate axes ?

But what if the new coordinate axis has different orientation i.e is offset by an angle with respect to the old one, is this what they mean by a different orientation ? In this case, the angle made by the vector with respect to the new coordinate axes cannot be the same as with respect to the previous coordinate axis right ? Doesn't it mean the vector has changed ?

 

[PeroK]

A vector is defined by its orientation in space. It's components relative to a chosen set of axes depend on the axes. This applies to polar coordinates as well. I.e. a vector is defined by $r, \theta$ relative to some axis. If you change the axis, then $\theta$ changes.

 

[Delta2]

The vector doesn't change, (neither the magnitude of a vector changes ), it is the basis of the vector space and the representation of the vector in that basis (in simple words, the vector components), that changes when we change a coordinate system.

If you know a bit of linear algebra, let's take a vector space $\mathbb{R}^3$ and a vector w=(1,1,1) of this space. This vector in the basis that consists of (1,0,0) (0,1,0),(0,0,1) has the components 1,1,1. If we change basis, the components will change and will be $\lambda_1,\lambda_2,\lambda_3$ but the vector will still be (1,1,1), and it will be $$(1,1,1)=\lambda_1 u_1+\lambda_2 u_2+\lambda_3 u_3$$ where $u_1,u_2,u_3$ the new basis.