Find the shortest distance between,

and

We can say that the vector perpendicular to the tangent vector of each curve, , is described as follows,

Now if I could get my two curves in the form of a position vector, I can differentiate both position vectors to obtain their tangent vectors.

Once I have their tangent vectors (Let's say vectors and ) I can cross them with my vector and equate them to 0.

But how do I get my curves in the form of a position vector?

And even after all this, I've only got 2 equations, so I need another one! Where can I find that last equation?

EDIT:

How do those look for position vectors of my curves?