Why does (ax, by) not transform like a vector under rotation?

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The discussion centers on why the expression (ax, by) is not considered a vector unless a equals b. The original poster attempts to demonstrate that this expression preserves length under rotation, suggesting it behaves like a vector. However, the response clarifies that the components ax and by transform differently under rotation because a and b are treated as scalars, leading to a failure in vector transformation properties. The key takeaway is that for (ax, by) to qualify as a vector, the coefficients a and b must be equal, ensuring consistent transformation under rotation. Understanding these transformation rules is crucial for distinguishing between vectors and non-vectors in physics.
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Homework Statement


Show that (ap1,bp2) is not a vector unless a = b.(The 1 and 2 are superscripts)
In Einstien's Gravity in a Nutshell, p 43, Zee states the above is not a vector because it doesn't transform like a vector under rotation. When I use the usual rotation matrix for rotation about the z axis (R11 = cosQ, R12 = sinQ, R21= -sinQ, R22 = cosQ), then check that length is preserved under this transformation, I get that this is in fact a vector.
Zee gives another example: (x2y, x3+y3). This, too seems to preserve length after being multiplied by the rotation matrix. What am I missing?
I would be happy to have an explanation for either example, of course.

Homework Equations

The Attempt at a Solution



I will use the first example, but make the notation easier by starting with the "vector" (non-vector?) (ax, by) = r.
r' = Rr = (axcosQ+bysinQ, -axsinQ + bycosQ)
squaring r' gets (axcosQ)2 + (bysinQ)2 +axbysinQcosQ + (-axsinQ)2 + (bycosQ)2 -axbysinQcosQ = (ax)2 + (by)2, which is r2, so length is preserved.

Seems pretty straightforward, so: say what?[/B]
 
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AMichaelson said:
(ax, by) = r.
r' = Rr = (axcosQ+bysinQ, -axsinQ + bycosQ)
When you write r' = Rr you are assuming that r is a vector. But you actually want to show that it is not a vector (if a ≠ b).

You are given that ##(x, y)## is a vector. So, ##(x', y')## is equal to the rotation matrix applied to ##(x, y)##. That is, you know how ##x## and ##y## transform when going to the primed frame.

When you write ##(ax, by)##, then ##x## and ##y## are still the components of the vector ##(x, y)##. ##a## and ##b## are assumed to be scalars; so, they don't change when going to the primed frame. Therefore, the quantity ##ax## transforms as ##ax## → ##ax'##. Similarly for ##by##. So, you can see how ##(ax, by)## transforms. Then you can check whether or not ##(ax, by)## transforms as a vector.
 
Thank you so much! (:doh: Of course!)
 
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