- #1
david.aloha
- 14
- 0
All right, I've been doing a lot of reading on quaternions, and while I think I understand how to use them, I'm still very confused as to why certain things are defined the way that they are. First question:
Why, when multiplying the imaginary portions of quaternions do you get these "rules":
i*i = j*j = k*k = -1
i*j = k, j*i = -k
j*k = i, k*j = -i
k*i = j, i*k = -j
I understand why i*i = -1, but why does an imaginary number times an imaginary number sometimes equal the negative of another imaginary number? I can't seem to find any underlying reason for this in anything I've read - it's just stated.
Second question:
Why use imaginary numbers at all if you're just defining a 3-vector plus a scalar (at least in terms of its use for 3D rotation which is the main context I've viewed it in)? What is complex or imaginary about it? The imaginary parts each define an axis in the same way as x, y, and z in a 3-vector - I'm just struggling to find what's different or special about adding the imaginary numbers (which, as I said above, also confuse me by only sometimes acting like imaginary numbers in multiplication).
I've spent a few days now looking through many different guides to quaternions, and none of them seem to answer these questions. Any help or clarification would be greatly appreciated. I'm the kind of guy who needs to know WHY something works, not just HOW it works, and this lack of why is driving me nuts.
Why, when multiplying the imaginary portions of quaternions do you get these "rules":
i*i = j*j = k*k = -1
i*j = k, j*i = -k
j*k = i, k*j = -i
k*i = j, i*k = -j
I understand why i*i = -1, but why does an imaginary number times an imaginary number sometimes equal the negative of another imaginary number? I can't seem to find any underlying reason for this in anything I've read - it's just stated.
Second question:
Why use imaginary numbers at all if you're just defining a 3-vector plus a scalar (at least in terms of its use for 3D rotation which is the main context I've viewed it in)? What is complex or imaginary about it? The imaginary parts each define an axis in the same way as x, y, and z in a 3-vector - I'm just struggling to find what's different or special about adding the imaginary numbers (which, as I said above, also confuse me by only sometimes acting like imaginary numbers in multiplication).
I've spent a few days now looking through many different guides to quaternions, and none of them seem to answer these questions. Any help or clarification would be greatly appreciated. I'm the kind of guy who needs to know WHY something works, not just HOW it works, and this lack of why is driving me nuts.