Books with a geometric approach to vectors? I'm taking multivariable calculus with a physicist, and I'm having a really hard time following the lectures. The problem is my lack of ability to picture vectors geometrically and interact with them in space. Does anyone know of some good books to study vectors geometrically?. This is what I have in mind: 


*

*Definition of vectors formally by means of euclidean geometry (aka directed segments)

*Definition of vector operations geometrically not analytically 

*Treatment of different types of vectors (arrows starting in different positions in space instead of all of them starting at the origin)

*The more formal, the better

 A: Andres Mejia has recommended a couple of books (Div, Grad, Curl is especially nice), but his books are for vector calculus and from your question it sounds like what you want (now, at least) are books for vector geometry.
Consider Elementary Vector Geometry by Seymour Schuster.
I got a copy of this (the original printing; it's been reprinted by Dover Publications) several years ago, and it seemed to me that it would serve nicely as supplementary reading for just what you are taking. In fact, a few years ago I was tutoring someone in an engineering multivariable calculus course, and I found this book helpful.
Also, see Calculus of Several Variables by Serge Lang.
Lang has a reputation for writing books that are concise with a lot left to the reader (for others reading, in 1978-79 I had a couple of graduate algebra out of Lang's Algebra, so I'm also speaking from personal observation), but this particular book is quite the opposite in that it is much more straightforward and reader friendly than the typical multivariable calculus text. Lang's book also has a very gentle treatment of vectors. For comments about Lang's book, see here (21 January 2011 ap-calculus post of mine archived at Math Forum) and Would it be fine to use Serge Lang's two Calculus books as textbooks for freshman as Maths major?.
A: This is not formal, but emphasizes a geometric intuition: Div, Grad, Curl.
Alternatively, there is a somewhat standard book, depending on what level of vector calculus you are looking for:
vector calculus
