- #1
proton
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Homework Statement
prove that quaternions are associative. ie (qr)s = q(rs), where q,r,s are quaternions
This isn't really a HW problem since I'm just wondering if there's a simpler way to prove associativity than the method I tried below
Homework Equations
i^2 = j^2 = k^2 = -1
ij=k=-ji
jk=i=-kj
ki=j=-ik
q=(a,b,c,d), r=(e,f,g,h), s=(l,m,n,p)
The Attempt at a Solution
Its not so much I don't know how to do the problem as it is that my work is tedious and will take forever, unless there's a simpler method
heres my work:
qr = (a,b,c,d)(e,f,g,h) = ae + af*i^2 + ag *ij + ah*k + be*i + bf*i^2 + bg*ij + bh*ik + ce*j + cf*ji + cg *j^2 + ch*jk + de*k + df*k + dg*jk + dh*k^2 = (ae-bf-cg-dh, af+be+ch-dg, ag-bh+ce+df, ah+bg=cf+de)
(qr)s = (ae-bf-cg-dh, af+be+ch-dg, ag-bh+ce+df, ah+bg=cf+de)(l,m,n,p) =
(lae-lbf-lcg-ldh-maf-mbe-mch+mdg-nag+nbh-nce-ndf-pah-pbg+pcf-pde, ...)
I got too tired to work out the i,j,k components
Theres got to be a simpler way to prove associativity