What is a good way to introduce vectors on a linear algebra course so that students are motivated from the start? I need an opening which will have a real impact. Are there any motivating examples?
You might like to take a look at an article on tensors by J. Kolecki from NASA: http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf. Although the article is primarily about tensors, he motivates them by motivating vectors in the introduction. He has a really nice way of explaining what scalars, vectors, and tensors are all about - all of my explanations to students are based off of his article.