Jump to content

  • Log In with Google      Sign In   
  • Create Account


Vector Multiplication


Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.

  • You cannot reply to this topic
1 reply to this topic

#1 CulDeVu   Members   -  Reputation: 560

Like
0Likes
Like

Posted 29 August 2012 - 09:14 PM

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
I'm sorry about any spelling or grammar mistakes or any undue brevity, as I'm most likely typing on my phone :P

Sponsor:

#2 apatriarca   Crossbones+   -  Reputation: 1620

Like
2Likes
Like

Posted 30 August 2012 - 03:23 AM

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.




Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.



PARTNERS