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

Show that every quaternion z, where |z|= 1, can be expressed

  1. Oct 13, 2009 #1
    hi, and thanks for reading. hh, and this isn't homework, its just something i've been wondering about.

    i've been flicking through a linear algebra book, i'm trying to learn it by myself, and i've come across this question which has completely stumped me:

    show that every quaternion z, where |z|= 1, can be expressed in the form z = cos(alpha/2) + sin(alpha/2).n, where n is a vector of length 1

    I don't know where to start, but more importantly, i don't understand the intuition behind it. Anybody care to explain? thanks
     
    Last edited: Oct 13, 2009
  2. jcsd
  3. Oct 13, 2009 #2

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Quaternions

    Write down a general quaternion z, and write down |z|.
     
  4. Oct 13, 2009 #3
    Re: Quaternions

    Thanks for the reply. I've that much done. And I know that a^2 + b^2 + c^2 + d^2 = 1. But, that's where I'm lost.
     
  5. Oct 13, 2009 #4

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Quaternions

    I assume that b, c, and d are the coefficients of i, j, and k respectively.

    What does

    a^2 + b^2 + c^2 + d^2 = 1

    and

    b^2 + c^2 + d^2 >= 0

    say about a^2, and thus about a?
     
  6. Oct 13, 2009 #5
    Re: Quaternions

    Innuendo, are you Irish?
    We got this exact same question for homework in Linear Algebra, to hand in today... /suspicious :smile:
     
  7. Oct 13, 2009 #6
    Re: Quaternions

    yes, b, c and d are the coefficients of i, j and k

    it says that a is less than or equal to 1?

    so, n = i + j + k, then b, c and d = sin(alpha/2)? and a = cos(alpha/2)? i can see that much, but i can't see how to get one from the other
     
    Last edited: Oct 13, 2009
  8. Oct 13, 2009 #7
    Re: Quaternions

    i've never been in a linear algebra class, im just working through problems that are in a linear algebra pdf i downloaded :)
     
  9. Oct 13, 2009 #8
    Re: Quaternions

    Ahh well if that's the case, then I can give you a few hints since I solved it myself.

    Think about how to construct [tex]\vec{n}[/tex] in such a way that satisfies the question.
    Think about when Sin/Cos is defined.
    Think about that equation George Jones gave you and there's a certain trig identity that may allow you to manipulate it.
     
  10. Oct 13, 2009 #9
    Re: Quaternions

    Hey Zorba I'm from Ireland and had to hand up this question in class today.
    You doin maths in trinity?

    As for the the question I couldn't quite get it.
    Sorry.
     
  11. Oct 22, 2009 #10
    Re: Quaternions

    Aye, I'm in Trinity, but doing TP though. :smile:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Show that every quaternion z, where |z|= 1, can be expressed
  1. Arithmetics in Z (Replies: 3)

Loading...