If omega bar is parallel to d r bar it is same as telling omega bar cross d r bar is 0, this is a cross product where d r where that d r is can be given by x i cap or d x i cap or d y j cap plus d z k cap. And already we know omega has a component, omega bar has a component omega x omega y and omega z. If that is the case then what will happen
omega bar d r bar, if that is 0, we can also find easily from this; that means, i j k you have omega x omega y omega z, these are the components of the vorticity vector