There are no planetary alignments in the next few decades, Earth will not cross the galactic plane in 2012, and even if these alignments were to occur, their effects on the Earth would be negligible.

Today, people easily cross the Atlantic Ocean by plane and cavalierly refer to the great body of water as "the pond."

Another way to do it, of course, would provide actually parametric equations of these lines, get vectors along them and then take the cross-product to get the normal vector to the plane.