Rotating a system

Question

brekely physics book chapter 2 page 30 , a question about rotating a system by $ \frac{\pi}{2} $ around the z axis clockwise direction and writing vectors according to the new axis after rotation

does the phrase rotate the axis always imply changing the reference frame we are using to describe the objects location and not rotate the object along side it since I had some trouble understanding the meaning,

as for how I solved I went and rotated the axis and reach a transformation matrix for the objects
$$ \left[ \begin {array}{cc} x'\\y' \\z' \end{array}\right]= \left[ \begin {array}{cc} 0 & 1 &0 \\ -1 &0&0& \\ 0&0&1\end {array} \right] \left[ \begin {array}{cc} x\\y \\z \end{array}\right] $$

secondly I want ask about the difference between applying the transformation to the vector itself vs to the basis, when I apply to the basis and then substitute the new basis in a a set of linear equations I can change the set of equations to a a new matrix and get the coefficients of the linear combination such that I have the new components according to the new basis

yet I noticed the matrix that I get there is the same as the one above there is some underlying confusion regarding this , I assume its related to the fact that every vector is a linear combination of the basis so when changing the it changes the coefficients too , I don't know why I got confused about this but certainly would appreciate something to fill the gaps ?

thirdly if some light can be shed on the difference between rotating the axis and rotating the vector itself while staying in the same coordinates system

Answer 1

As for what most people mean when they say rotate the axis - yes, it usually implies keeping all objects in the same points in space, and just using new coordinates to express their position, velocity etc.

Regarding your second question, I didn't quite understand what you meant, but if I did, what you're asking is essentially "what's the difference between transforming the vector or transforming the original base?" I don't think there is any difference. To obtain the matrix which you use to directly transform the vector, you would need to know how your base behaves under the transformation and build the transformation matrix.

About the difference between rotating the axis and rotating the vector: Rotating a vector is very similar to rotating the entire set of axis.Consider the vector $\begin{bmatrix} 1 \cr 1 \end{bmatrix}$ being rotated $45^\circ$ clockwise (making it the vector $\begin{bmatrix} 0 \cr 1 \end{bmatrix}$. Instead of doing this, we could've rotated the entire [xy] plane $45^\circ$ counter-clockwise, achieving the same result, coordinates wise. Note that many times we have more than one object in a system, and so when rotating one vector, we would have to put all vector through a similar process (to solve the same problem we had in the first place, e.g. two rubber balls colliding - if we only rotate one's velocity, they won't collide anymore) which would be terribly long, so it's generally much more preferable to rotate the axis instead.