Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is the largest number of mutually obtuse vectors in Rn?

  1. Jan 28, 2016 #1
    This is my question:

    What is the largest m such that there exist v1, ... ,vm ∈ ℝn such that for all i and j, if 1 ≤ i < j ≤ m, then ≤ vivj = 0

    I found a couple of solutions online.
    http://math.stanford.edu/~akshay/math113/hw7.pdf (problem 10. But it's basically the same solution as the one in the link above)

    I kind of understand the contradiction but I don't get why there won't be a contradiction when you take m = n + 1. This is my first course in linear algebra, so far I've learnt about linear independence, subspaces, orthogonality but I'm not very familiar with things like inner product spaces - only dot products. I need someone to explain this in simpler terms.
  2. jcsd
  3. Jan 28, 2016 #2


    User Avatar
    Science Advisor
    Homework Helper

    I'm referring to the argument in the pdf.

    In case b), where M or N is 0, they use the last, until then unused, vector, ##v_{n+2}##, to get the contradiction. So we need one vector kept on the side.

    Now let's go back to the start of the proof. If m=n+1, and we have to keep one vector on the side, there are only n vectors left to work with. But the argument started with noticing that the n+1 vectors must be linearly dependent. That worked because we had n+1 vectors in an n-dimensional space. With n vectors in an n-dimensional space, you can't be sure that they are linearly dependent. Hence, you don't have the equation marked by (*) in the pdf.
  4. Jan 28, 2016 #3
    I got it. Thank you so much.
  5. Jan 28, 2016 #4


    User Avatar
    2017 Award

    Staff: Mentor

    It is interesting that the question is quite easy to answer for 90 degrees, but an unsolved problem (Kissing number problem) for 60 degrees.
  6. Jan 28, 2016 #5


    User Avatar
    Science Advisor
    Homework Helper

    as often happens your problem is incorrectly stated here. you have to specify non zero vectors to get a contradiction. i.e. a sequence of zero vectors, no matter how long, satisfies the statement you gave above.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook