If you're looking for a higher level understanding of vectors and vector application. There is a book called Div, Grad, Curl. I can't remeber the author. There are multiple editions. Most larger libraries will carry a copy of at least one of the editions. The first chapter covers vector...
Have you tried assuming that there are two solutions (just label as s_1 and s_2). Then apply them through the equations and find a contradiction to the assumption.