Question

What is the connection between cross product and quaternions?
Is that where cross and dot product come from?

Answer 1

This is my guess..
if you look at the quaternion wiki page there is a small graph which shows the commutivity of quaternions ( http://en.wikipedia.org/wiki/Quaternion)

Now if you construct a 3x3 matrix with i,j,k at the top you will see that you find your cross product values are clearly what helped derive this matrix. I.e if you check the determinate under the j column you will get -j = i * k just as shown in the graph. This holds true for all the component cross products. However im not sure where the 1 came from.