Quaternion division not equal to multiplication with inverse (q1 / q2 != q1 * q2^-1)?

Question

From several docs, like [1] and [2], I learned that divide operation on a quaternion is equivalent to multiply its inverse. That is to say, for two quaternions q1 and q2, we have

However, when I verify this in MATLAB, which will give different results (also verified by further converting them to rotation matrix via quat2rotm). See code below:

q1 = [1 0 1 0];
q2 = [1 0.5 0.5 0.75];
q1 = quatnormalize(q1);  % this seems doesn't matter
q2 = quatnormalize(q2);  % this seems doesn't matter

res_1 = quatdivide(q1, q2)               % this will be [0.7385  0.1231 0.2462 -0.6155]
res_2 = quatmultiply(q1, quatinv(q2))    % this will be [0.7385 -0.6155 0.2462 -0.1231]

Any hint is appreciated.

Answer 1

Note that quaternion division leads to an ambiguous notation.

reading from your 2nd reference it states

To expand upon this see what the result of the two interpretations are:

They differ by the sign of the cross product in the vector part.

I suspect that quatdivide() uses the 2nd convention such as

quatdivide(q1,q2) = quatmultiply(quatinverse(q2),q1)