1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Verify that it's a vector

  1. Jan 5, 2017 #1
    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?
     
  2. jcsd
  3. Jan 6, 2017 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Welcome to PF!

    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.
     
  4. Jan 13, 2017 #3
    Thank you so much! (:doh: Of course!)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Verify that it's a vector
  1. Someone verify? (Replies: 8)

  2. Verify this concept? (Replies: 8)

Loading...