Need help with Fluid mechanics using mathcad symbolic calculation

Please see attached for the problem I am doing.

I think I am making things too complicated.

In the material description of a fluid particle, the vorticity tensor in involved.

First I do not understand how the dimension is going to work out when the question has a sin( product of tensor and time) ...

Second, when I want to define the material description implicitly, I have the vorticity tensor in the way and it make things look very complicated.

Thanks