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[CulDeVu]

Hey everyone!

I was learning about quaternions the other day and I noticed something interesting about quaternion multiplication.

The component-wise form of Q3=Q1*Q2 is
w3=w1*w2 - v1.v2
v3 = v1*w2 + v2*w1 + v1Xv2

I noticed it looked very similar to a simple algebraic expansion of the form (a + b)(c + d).

Well if you look at the multiplication as a multiplication of their components you get:

(w1 + v1)(w2 + v2) = w1*w2 + w1*v2 + w2*v1 + (v1)(v2)

If you cancel terms from the first set of equations:

(v1)(v2) = v1Xv2 - v1.v2

What type of multiplication is this?

Thanks,
CulDeVu

[apatriarca]

If you write the quaternions as 4D vectors with basis 1, i, j, k, then the multiplication is simply defined using the distributive law you used in your post with rules i^2 = j^2 = k^2 = -1 and i*j = k, j*k = i, k*i = j (the other products follows from these identities). Since the scalar-vector representation of the quaternions is equivalent to this representation, it is clear that the product in that representation should resemble the application of the distributive law. That product is simply the quaternion product of two imaginary quaternions. The result is indeed a non-imaginary quaternion having both a scalar and vector component.