Intuition behind quaternion multiplication with zero scalar Grouping into vector notation two quaternions
$$(a,b,c,d)= a + b \mathbf i + c\mathbf j + d\mathbf k=(a, \vec x)$$
and
$$(e,f,g,h)= e + f \mathbf i + g\mathbf j + h\mathbf k=(e, \vec y),$$
their multiplication
$$(a,\vec x) \cdot (e,\vec y)=(ae-\vec x \cdot \vec y, a\vec y+ e \vec x + \vec x \times \vec y).$$
If the scalar components of each quaternion are zero,
$$(a,\vec x) \cdot (e,\vec y)=(-\vec x \cdot \vec y,  \vec x \times \vec y).$$
This result is the vector formed by the negative of the dot product, $-\vec x \cdot \vec y,$ as its first element, and the cross product, $\vec x \times \vec y,$ as its second element.  
The question is whether there is a physical, physics, geometrical or vectorial intuition of this result with the idea of helping in the understanding of quaternions.
 A: If both vectors $x$ and $y$ are of unit length, the resulting quaternion is actually a rotation around the axis formed by the cross product of the two vectors $x \times y$.  Since quaterions are applied in "sandwitch" product,  the rotation of any vector around that axis is twice the angle formed by the two vectors. The rotation angle of the quaternion can be calculated as:
$\theta = \tan^{-1}(\frac{\| x \times y\|}{x \cdot y})$.
The quaternion product of $x$ and $y$ is equal to:
$x y = -\cos(\theta) + \sin(\theta) b$
Where $b = \frac{x \times y}{\|x \times y\|}$ is the axis of rotation.
That is analogous to a 2D rotation in complex number theory (Euler's formula)
$e^{-\theta i} = \cos(\theta) - \sin(\theta) i$
You can also define a quaternion in terms of exponential function:
$e^{-b \theta/2} = \cos(\theta/2) - \sin(\theta/2) b$
The convention here is to use half of the angle, since quaterions are applied in "sandwitch" product, the total rotation is twice the angle of the quaternion. 
Using the exponential map defined above, one can parameterize quaternions using angle and axis parameters.
If vectors $x$ and $y$ are not of unit length, the quaternion $q = x y$ is still representing a rotation, since the inverse of $q^{-1} = \frac{q^*}{\|q\|^2}$,  the scalar factor $\frac{1}{\|q\|^2}$ accounts for the normalization of the rotated vector $v' = q v q^{-1}$. So the norm of $v$ is preserved after rotation.
Quaternion product is a form of the fundamental "geometric product" of vectors in the 3D Euclidean Geometric Algebra. That product convey a lot of geometric meaning. In the language of Geometric Algebra you can define projections, rejections, intersections and more, in terms of the geometric product.
