This is taken from the famous 3 polarisation filter experiment on light and got my thinking about coordinate transformations. and vector component consistancy and would really like an answer.
Suppose in the regular coordinate system a vector which is 0i+yj obviously has a magnitude y
Now, lets say I switch coordinates such that my new axis are 45 degrees rotated anticlockwise. with basis vectors i' and j'
my vector is simply ycos(45)i'+ysin(45)j'
now this is where the physics comes in. lets say something gets rid of this j' component of the vector( acting as a polarisation filter)
my new vector is now simply ycos(45)i'
now... when we switch back to my original coordinates. this vector now has an i component and a j component...
which if im not mistaken will be ycos^2(45)i + ycos(45)*sin(45)j
So my question is. how can it be that by switching coordinates. and REMOVING a component in a different frame, not adding anything to it at all. and then switching back to our original coordinates we have magically produced an i component when it never had any before?
this works with light and im totally unsure about how this makes any sense
One metaphorical way to grasp it is that the filter changes the vector. You could imagine it as rotating and shrinking the vector, instead of just cutting it and removing one component.
Another way to think about it is that the filter removes the $j'$ component, which is like subtracting that much of the vector. But if you start with zero and subtract you don't get zero. That's why you can end up with an $x$ component, because the thing you subtracted had an $x$ component.
It comes down to precisely describing "get rid of the $j'$ component." I have given two ways that might be done. There are surely others as well.