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Quaternions
Name: bryan a bertoglio
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
Could someone explain quaterions to me? When working with fractals, I
noticed a puzzling set of rules that they follow:
1: i^2 = j^2 = k^2 = l^2 = -1
2: sorry, I forgot, but I think i * j = -j * i or something weird
It had to do with plotting fractals in these four dimensions, fixing one, a
and making 3-D fractals. I could not picture any way the rules could work,
and my calculus teacher had never heard of them. If you need the other
rule, I will post it later (x*y is not equal to y*x for these . . . how?)
Replies:
The system of quaternions is an extension of the complex number system in
the same way that the complex number system is an extension of the real
number system. Recall that you treat 2 + 3i algebraically just as you would
treat
2 + 3x except that when multiplying you replace any occurrence of i*i by -1.
You would treat a quaternion such as 2 + 3i + 5j - 2k algebraically just as
you would treat 2 + 3x + 5y - 2z except that you replace products of i, j,
and k according to the following rules:
i * i = -1, j * j = -1, k * k = -1, i * j = k, j * k = i, k * i = j,
j * i = -k, k * j = -i, and i * k = -j.
As you observed, this does make multiplication not commutative which may
seem weird, but is not any weirder than the i * i = -1 in the complex
numbers or the non-commutativity of matrix multiplication.
chaffer
Also, if you are familiar with cross products and dot products, if you
represent a 3-dimensional vector (x, y, z) by the
quaternion x i + y j + z k, then the product of two such 3-D vectors, using
the quaternion multiplication rules gives a quaternion A i + B j + C k + D,
where the 3-D vector (A,B,C) is the cross product of the two original
vectors, and D is the negative of their dot product. So quaternion multi-
plication actually is slightly related to three-dimensional vector opera-
tions.
asmith
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