Title here, "From Seattle move in a straight line due East. What country will you hit?" is subtly different from what the article is describing: "From Seattle, face due East. Now, continue forward in a straight line..."
In my mind, "move in a straight line due east" can be interpreted as "for as long as you are moving, your movement should be due east". In that case, your latitude will never change. In the northern hemisphere, you will constantly be making a leftward adjustment to maintain a due east heading.