Vector calculus

1. The problem statement, all variables and given/known data

A vector field $$ \vec{u}=(u_1,u_2,u_3) $$

satisfies the equations;

$$ \Omega\hat{z} \times \vec{u}=-\nabla p , \nabla \bullet \vec{u}=0$$

where p is a scalar variable, [itex] \Omega [/itex] is a scalar constant. Show that [itex] \vec{u} [/itex] is independant of z.

Hint ; how can we remove p from the equations



2. Relevant equations

Included above in question.





3. The attempt at a solution

I know that it means that [itex] \vec{u} [/itex] doesnt have a z component and therefore is only described by x,y but I have no idea where to begin.

I tried removing p but I can't.



[edit]

I have made some progress I took the curl of the longer equation and got rid of [itex] \nabla p [/itex] using the curl of a scalar gradient = 0, but from then I just have ;

[itex] \nabla \times ( \vec{u} \times \Omega \hat{z})=0 [/itex]