Maths - Vector Expansion - Forum Discussion

from: nookabooka
2009-10-03 02:24:04 UTC

Hello all, not sure if this is the right place to post about this, but here goes.

Im working through the details of deriving the rotation vector/operator that is then put in matrix form on this website:- https://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/transforms/derivations/vecExp/index.htm

Once I get to this part:- (-sq(vp•vq) + vq•(-vp x vq + sqvp) , - vq x (-vp x vq + sqvp) + sq(-vp x vq + sqvp) - vq(vp•vq))

I am confused on how to simplify the real part of the quaternion down to - vq•(vp x vq)

I suppose Im confused on how to simplify any part of it :P but one bit at a time.

thanks


from: martinbaker
2009-10-03 08:22:42 UTC

we want to simplify:

-sq*(vp•vq) + vq•(-vp x vq + sq * vp)

expanding out the second part gives:

-sq*(vp•vq) - vq•(vp x vq) + vq•(sq * vp)

the scalar part can be moved outside the brackets: vq•(sq * vp)=sq*(vq•vp)

unlike the cross product the dot product commutes so: vq•vp=vp•vq so we get:

-sq*(vp•vq) - vq•(vp x vq) + sq *(vp•vq)

canceling out the first and last terms leaves the middle term: - vq•(vp x vq)

On the website I have explained things in terms of scalar and vector (s,v) as well as the more usual notation: a + i b + j c + k d this is because different people seem to latch onto different ways of understanding it. So if you find this approach difficult, I would try the other notations, I don't think it is necessary to understand all approaches to use quaternions or understand quaternion, matrix conversions.

Martin

 

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