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 Quaternions 
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Joined: 25 Feb 2008, 13:32
Posts: 106
Location: Cape Town, South Africa
Post Quaternions
So the following relations are suppose to follow from i^2=j^2=k^2=ijk=-1:

ij=-ji
jk=-kj
ki=-ik


But, for instance, multiplying through by i in the relation ij=-ji yields:

i^2j=-ji^2
-j=j


This is my problem, how can -j=j . I suspect that I've missed some vital detail, perhaps some subtly to do with the commutation law.

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26 May 2008, 02:55

Joined: 12 Mar 2008, 10:57
Posts: 69
Location: India
Post Re: Quaternions
Multiplication in quatenionic algebra should be considered an "operation" for conceptual clarity.(e.g multiplication by iota in the complex plane signifies an anticlockwise rotation by a right angle)
Suppose we have
ij=-ji
When the operator i acts on ij the result will be the same as i acting on -ji.That is,
i(ij)=i(-ji)=-iji(this is called pre multiplying by i)
Now we know,
i=i
Pre multiplying b/s by ij we have
ij(i)=ij(i)or
iji=-ji(i)(because ij and -ji are the same operators)
This is called post-multiplication by i.
You have pre-multiplied by i on the LHS but post-multiplied on the RHS which has led to this error.


26 May 2008, 09:16
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Joined: 25 Feb 2008, 13:32
Posts: 106
Location: Cape Town, South Africa
Post Re: Quaternions
Thank you Sameed. The ordering of symbols is essential. Viewing these entities as "operators" makes this clearer. I have many, many more questions; too many to ask here though - I'll have to save up to buy a book on quaternions (and octonions).

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29 May 2008, 16:17
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