Yakk

-1 = i^2 = j^2 = k^2
i = jk = - kj
j = ki = -ik
k = ij = -ji

Edit2: The above 4 lines are all you need to know. What follows are mnemonics and tricks and other ways to help you remember the above 4 lines.

start with:
i j k
and look at the identies:
i=j k
_ j=k i
(I rotated the i around from the left to the far right)
_ _ k=i j
(I rotated the i and j around from the left to the far right)

any "base" element (i j or k) is equal to the product, in order, of the "next two".

Any two different "base" elements are negated when you multiply them together the opposite way.

Any "base" element squared is negative one.

Edit:
Some more math that might make it make more sense.
if x is i j or k, then:
x^2 / x = -1 / x
x = - 1/x
1/x = -x
so
1/i = -i
1/j = -j
1/k = -k
(or, in english, the multiplicative inverse of i is negative i)

start with
i = j k (eqn 1)
divide by k on the right:
i / k = j k / k
i * -k = j
- i k = j
j = - i k (eqn 2)

divide eqn 1 by j on the left: (same as multiplying by -j)
1/j * i = 1/j * j * k
1/j * i = k
-j * i = k
k = - j i

Multiply eqn 2 by j:
-1 = j^2 = - i k j = - j i k
i k j = j i k = 1
if
i k j = 1
then
-i * i * k j = -i
i = -k j (eqn 3)
and you can use eqn 3 to prove the other two identities.

Not sure if this makes things make more sense. =)

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Last edited by JHVH : 10-29-4004 BC at 09:00 PM. Reason: Time for a rest.