The product between quaternion

[HeilPhysicsPhysics]

For example:
i^2=?
j^2=?
k^2=?
ij=?
jk=?
ik=?
ijk=?
Is ij=ji?
And how to prove them?

And also,vector times vector, what is the product?

 

[d_leet]

For example:
i^2=?
j^2=?
k^2=?
ij=?
jk=?
ik=?
ijk=?
Is ij=ji?
And how to prove them?

And also,vector times vector, what is the product?

This is from wikipedia http://en.wikipedia.org/wiki/Quaternions

i² = j² = k² = ijk = -1
ij = k
jk = i
ki = j
ji = -k
kj = -i
ik = -j

The first one I think is the basic definition and the rest follow from that, some of the proofs are on the wikipedia page, and what do you mean by a vector times a vector? The above equalities allow a general definition of a product of quaternions.

 

[HeilPhysicsPhysics]

Just like:
A=i+j+k
B=2i+3j-4k
They are both vector.
I heard that a vector times a vector become a tensor(rank two).
If ij=-k,ik=-j,jk=-i.
The product is:
(i+j+k)(2i+3j-4k)=-2-3k+4j-2k-3+4i-2j-3i+4=i+2j-5k-1
I think it is still a vector, not a tensor(rank two).

 

[Data]

The product of two quaternions is still a quaternion (and quaternions are not vectors). You did the multiplication wrong up there, if A and B are as you had then AB is -7i + 6j + k - 1.

You can define many products between vectors, for example in $\mathbb{R}^3$ you have the usual cross and dot products (and the dot product generalizes to other spaces of course). Those two products can be read off the result of quaternion multiplication:

$$<1,1,1> \times <2,3,4> = <-7,6,1>$$

and

$$<1,1,1> \cdot <2,3,4> = 1 = -(-1).$$

In general if the product of two quaternions $A = a_1 i + a_2 j + a_3 k$ and $B = b_1 i + b_2 j + b_3 k$ is $AB = C = c_1 i + c_2 j + c_3 k - c_4$, then $<a_1, a_2, a_3> \times <b_1, b_2, b_3> = <c_1, c_2, c_3>$ and $<a_1, a_2, a_3> \cdot <b_1, b_2, b_3> = c_4.$