Why is it that for quaternions, $u*v = \mathrm{cross}(u,v)-\mathrm{dot}(u,v)$?
I wonder for what reason they are equal to each others.
Why is it that for quaternions, $u*v = \mathrm{cross}(u,v)-\mathrm{dot}(u,v)$?
I wonder for what reason they are equal to each others.
Assume that $u = bi + cj + dk$ and $v = xi + yj + zk$ are imaginary quaternions (no real part). Then a straightforward computation using the identities $i^2 = j^2 = k^2 = ijk = -1$ gives \[ uv = -(bx + cy + dz) + (cz - dy)i + (dx - bz)j + (by - cx)k = - \langle u,v \rangle + u \times v \] with the usual identification $\mathbb{R}^{3} = \operatorname{Im}\mathbb{H}$.
The quaternions written are called "pure quaternions", meaning the scalar value is zero. Let me write a quaternion as a scalar and a 3-vector, where the 3-vector has an arrow. Then: $(0, \vec{u})(0, \vec{v}) = (-u \cdot v, \vec{u} \times \vec{v})$ This is not very general because for a different inertial observer, the scalar will no longer be zero. In that case: (a', \vec{u'})(b', \vec{v'}) = (a' b'- \vec{u'} \cdot \vec{v'}, a' \vec{v'} + b' \vec{u'} + \vec{u'} \times \vec{v'}) If the 3-vectors represent a position in space, then the scalars are time. If the 3-vectors are 3-momentums, then the scalars are energy.