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I just wanted to begin learning 'Vectors'. But I am completely confused where I would have to start! There is vector Algebra, Vector geometry, Vector Analysis and what not.

So, I want to know what are all the most widely used branches of vectors in existence.

PS:- I am a 'wanna-be' Physicist. I just realized vectors are so important for Physics and so I want to know it. So I would also be obliged if you could tell where a noob would have to begin with his study of vectors. I would be heavily indebted to you if you could also suggest a free downloadable ebook on the subject. (I suppose my last 2 requests are not so on topic and so you can answer to those only if you wish to!)

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  • $\begingroup$ I suggest you start here: en.wikipedia.org/wiki/Vector_%28mathematics_and_physics%29 $\endgroup$ – user_of_math Jul 19 '14 at 5:45
  • $\begingroup$ Have you checked Wikipedia to see what each of them do? In any case, linear algebra is where to start. $\endgroup$ – Gina Jul 19 '14 at 5:47
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    $\begingroup$ It is after referring to several Wikipedia articles that i got confused! $\endgroup$ – VenkiPhy6 Jul 19 '14 at 5:53
  • $\begingroup$ A lot of people first learn about vectors in a vector calculus course or in a freshman physics course. Vectors are also introduced sometimes in a pre-calculus class or in high school algebra. Perhaps this Khan Academy video is one place to start. $\endgroup$ – littleO Jul 19 '14 at 7:01
  • $\begingroup$ I was actually browsing other questions on this site for Pre calculus books when you posted that comment @littleO ... nice coincidence! $\endgroup$ – VenkiPhy6 Jul 19 '14 at 7:02
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In the first place, a vector is an element of a vector space.

A vector space is an environment (a set) where it makes sense

  • to add any two elements $x$ and $y$, producing $x+y$,

and

  • to scale any element $x$ with arbitrary real (or complex) scalars $\lambda$, producing $\lambda x$,

such that reasonable rules, like $$1\> x=x,\quad x+y=y+x,\quad \lambda(x+y)=\lambda\>x+\lambda\>y,\quad{\rm etc.},$$ hold.

Given that vector space is such a general notion there are all sorts of environments that have the structure of a vector space, and all sorts of things that can be called "vectors": triples, polynomials, continuous functions $f:\ [0,1]\to{\mathbb R}$, translations of euclidean $d$-space, equivalence classes of directed segments $[a,b]$, what have you.

Many of the vector spaces appearing in mathematics have additional structure elements and corresponding qualifying names: a norm ("normed space"), a scalar product ("euclidean space"), a "cross product" (euclidean $3$-space), a "pointwise product" (function spaces), a "convolution", and so on.

The algebraic, geometric, and computational theorems valid in "finitely generated" (= finite-dimensional) vector spaces are dealt with in linear algebra, the laws valid in connection with "vector-valued functions" and differentiation are dealt with in vector analysis, and so on.

What I'm saying is: There is no single book telling it all about vectors. "Vectors" are a façon de parler. The essential thing is that you intuitively accept a thing being a "vector" when told to do so.

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    $\begingroup$ Okay, so the concept of vectors is applicable to several math fields and providing an exhaustive list of all the fields it is applicable to, in a single answer is not so easy. Probably this also means a strong knowledge of vectors might be useful in several fields of math, letting one to have a strong knowledge of math itself. Am I right? If I am, please just up vote this comment and that will do. Thanks BTW. $\endgroup$ – VenkiPhy6 Jul 20 '14 at 14:35
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A good start would be to pick up a good linear algebra book. Euclidean vectors, which is what you'll first see in physics as being called "vectors", are only one way of treating vectors. The concept is much more general and abstract, and much more powerful (as I joked with this parody of Magritte's "The Treachery of Images")

The choice of literature depends on your level. A good introduction is "Introduction to Linear Algebra" by Gilbert Strang, who also has his MIT lectures available at MIT Open Courseware. This is a a more practical and intuitive take on the subject which I think is a good thing.

Serge Lang's "Linear Algebra" is a very common linear algebra textbook in undergraduate courses, and it's a good comprehensive book for the core theorems with a more formal and abstract take. To supplement the formalism, I suggest taking up "Linear Algebra Done Right" by Sheldon Axler.

With that formalism in your belt, then you can start seeing vectors as a more general tool, something that will be very important later on if you decide to study physics formally and study analytical and quantum mechanics.

Good luck!

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    $\begingroup$ I agree with most aspects of your post except one. I would never recommend any book by Serge Lang to any beginner. I'm not sure I can forgive you for your crime =P $\endgroup$ – Christopher A. Wong Jul 19 '14 at 5:53
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    $\begingroup$ @LucasVB What about Gilbert Strang's Linear Algebra and its applications along with his Introduction to linear algebra? Maybe with these two I will get both the abstract and application orientd take on vectors? $\endgroup$ – VenkiPhy6 Jul 19 '14 at 6:36
  • $\begingroup$ I never checked it out, so I cannot comment on it. It seems to be a bit more advanced. I liked Introduction since it is an easy and well explained introductory text, and offers a lot of good and accurate intuitions about vectors and matrices, something I haven't found in any other text. If his other book has the same approach to the topic, then I would definitely recommend it. $\endgroup$ – LucasVB Jul 19 '14 at 6:43
  • $\begingroup$ So at last I bought 'Introduction to linear algebra' by Gilbert Strang as you said! Thanks! $\endgroup$ – VenkiPhy6 Jul 22 '14 at 18:19

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