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>I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this.
Think it is right.
>>
>> Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j.
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>Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right.
They are not the same thing. All of these are equal to -1 when squared, but they are different when multiplied by any other of this set and multiplication between them is anti-commutative.
>As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason.
>
These are not added for unknown reasons. They specify rotations of unitary vectors of a canonical base around one another.
>> Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing:
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>> pqp'
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>> where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u .
>>
>Oooooh kay... 1) Where'd w go?
The coordinate 'w' is the real part. For the vector 'u', such coordinate is 0. For 'p' and 'p'', it is cos(theta/2).
>Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)?
No.
>Would we need to bring it back in to apply the rest of this?
No. Just follow the rules to multiply hypercomplex numbers and the rotation will occur.
>2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?,
The notation '(x, y, z)' is just a shorter notation for "xi + yj + zk". For this problem, the vector 'u' defines the axis of rotation.
>3) A quaternion is... a point, then? Since we're rotating around it?
You can see a quaternion as a point in R^4 since it has 4 coordinates. Actually it describes an axis of rotation, by its imaginary part, and the angle of rotation.
>4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it?
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The axis of rotation is actually the vector "u".
