I'm a little rusty at this, so if anyone wants to correct my logic, please do so.

The cross-product gives a vector (in this case u), which is orthogonal to both v and w.

i.e. u is normal to the plane that v and w lie on, and also has the same coefficients as that plane.

So find the plane that v and w lie on, and then you have the normal, and so you have the cross product.