The question: Is the multiplication result of two unit-length quaternions "perpendicular" to both operands? "Perpendicular" means that dot product between either operand and multiplication result will be zero (as with vector3 cross-product). Operands (say q1 and q2) are not equal to each other.
Reason for asking: I'm trying to understand quaternion rotations from purely geometric point of view (even if it is 4d). For example, 3x3 affine transform matrix in 3d space can be represented as a new coordinate system into which a point should be placed, and each row(or column - depending on notation) of such matrix will represent a vector of a coordinate axis of the "new" coordinate system. So to transform a point with matrix, you simply take every vector that represents axis of a "new" coordinate system, scale it using numeric value of corresponding x/y/z of the point to transform, and add them all together to get new point. Such representation is very easy to understand. I'm looking for similar explanation of quaternions, and If I get confirmation that result of quaternion multiplication is "perpendicular" to both operands that would be at least something to get started.