1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. 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?