1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Quaternions multiplication

  1. Jun 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Its not really homework problem, and you may find it silly because its only multiplication problem, but I cannot get the right answer by multiplying quaternions.

    Basically this is what i want to show:

    exp(iψ/2)exp(kθ/2)exp(iф/2) = cos(θ/2)exp(i[ψ+ф]/2) + ksin(θ/2)exp(i[ψ-ф]/2)

    2. Relevant equations

    3. The attempt at a solution

    I begin writing:
    exp(kθ/2) = cos(θ/2) + ksin(θ/2)

    Then multiplying:
    exp(iψ/2)*[cos(θ/2) + ksin(θ/2)] = exp(iψ/2)cos(θ/2) + exp(iψ/2)*ksin(θ/2) =
    = cos(θ/2)exp(iψ/2) + sin(θ/2) exp(iψ/2)*k

    Since scalar terms can are commutative in quaternions algebra.

    Finally multiplying answer above with the final exp(iф/2)

    [cos(θ/2)exp(iψ/2) + sin(θ/2) exp(iψ/2)*k] * exp(iф/2) =
    = cos(θ/2)exp(iψ/2)exp(iф/2) + sin(θ/2)exp(iψ/2)*k*exp(iф/2) =
    = cos(θ/2)exp(i(ψ+ф)/2) + sin(θ/2)*k*exp(-iψ/2)*exp(iф/2) =
    = cos(θ/2)exp(i(ψ+ф)/2) + k*sin(θ/2)*exp(i(ф-ψ)/2)

    Here I used that exp(iψ/2)*k = k*exp(-iψ/2)

    And no matter how I do it I always get the same answer, with the last exponential term having ф-ψ, and the paper says it should be ψ-ф
  2. jcsd
  3. Jun 27, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    hi amiras! :smile:

    yours looks correct to me …


    = exp(iψ/2)cos(θ/2)exp(iф/2) + exp(iψ/2)ksin(θ/2)exp(iф/2)

    = cos(θ/2)exp(i(ψ+ф)/2) + kexp(-iψ/2)sin(θ/2)exp(iф/2)
  4. Jun 27, 2012 #3
    Thanks for confirming me, from now on I'l stop blindly following and start thinking by myself. :)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Quaternions multiplication Date
Question in quaternion multiplication Jan 12, 2015
Quaternions and Clifford Algebra problems Nov 10, 2014
Quaternion group Dec 12, 2013
Homomorphisms of Quaternion Group Oct 27, 2013
Multiplication of quaternions Apr 9, 2007