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!

Homework Help: Help with norms

  1. Apr 11, 2010 #1
    If [tex]\lVert x \rVert=2[/tex] and [tex]\lVert y \rVert=3[/tex], what if anything, can we conclude about the possible values of [tex]\left\vert \mathbf{x}^T\mathbf{y} \right\vert[/tex]?

    I don't think anything can be concluded since the dot product can still end being positive or negative.
     
  2. jcsd
  3. Apr 11, 2010 #2
    Re: Norms

    Do you know the formula for the dot product involving cosine?
     
  4. Apr 11, 2010 #3

    Mark44

    Staff: Mentor

    Re: Norms

    x [itex]\cdot[/itex] y = ||x|| ||y|| cos([itex]\theta[/itex]).

    Can you conclude something about |x [itex]\cdot[/itex] y| now?
     
  5. Apr 11, 2010 #4
    Re: Norms

    [tex]u \cdot v=\lVert v \rVert\lVert u \rVert cos(\theta)[/tex]
     
  6. Apr 11, 2010 #5
    Re: Norms

    It is between 0 and 1, then?
     
    Last edited: Apr 11, 2010
  7. Apr 11, 2010 #6
    Re: Norms

    Theta is between -pi/2 and pi/2?
     
  8. Apr 11, 2010 #7
    Re: Norms

    Plug in all the values you know. Then consider the range of cosine. What values can it take? Knowing this, what values can ||x|| ||y|| cos theta take?
     
  9. Apr 11, 2010 #8
    Re: Norms

    For cosine to be positive, theta is between, and including, -pi/2 to pi/2. Therefore, the right side of equation will be between 0 to 1 times the magnitude of x times the magnitude of y?
     
  10. Apr 11, 2010 #9
    Re: Norms

    Yes, it is simpler to just write it out, however.

    xy = ||x|| ||y|| cos a = 6 cos a.
    cos a is between -1 and 1, so xy is in [-6, 6], and so |xy| is just the positive terms in that interval.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook