Quaternion (0,0,0,0) signifies what

Question

After performing quaternion multiplication I am getting (0,0,0,0). Could anyone let me know what it signifies.

(1 0 0 0) - Identity quaternion, no rotation,
(0 1 0 0) - 180° turn around X axis,
(0 0 1 0) - 180° turn around Y axis,
(0 0 0 1) - 180° turn around Z axis,

Likewise what could be the explanation for (0,0,0,0)

Solution 1

As mentioned by John Coleman, the (0,0,0,0) quaternion is the zero element. The only way to obtain this result using multiplications is to multiply by (0,0,0,0). Since we don't not much about the context, we cannot be more specific about the origin of the problem.

About the meaning of the (0,0,0,0): geometrically, a quaternion encodes a rotation AND a scale transformation: q = s*u, where 'q' is the quaternion, 's' is a scaling factor (a factor 's' leads to a s^2 scaling), and 'u' is a unit quaternion (a rotation). Since 'u' is non-zero by definition, the only way to obtain (0,0,0,0) is to have a scaling factor of zero combined with an undefined rotation.

Solution 2

(0,0,0,0) is not a valid rotation quaternion, as they are required to have length 1.

If you want to get an intuition and some background information on how quaternions can describe 3d rotations, this is the perfect starting point:

https://eater.net/quaternions

The two guys did an outstanding job explaining and visualizing quaternions.