About Vectors (Dover Books on Mathematics)

About Vectors begins by setting forth the goal of defining vectors. The author then shows that there are problems with any definition of vectors. For example when are two vectors equal? When they have the same magnitude and direction, but what if one of those vectors is a free vector and the other a bound vector? After raising questions about what constitutes a vector the author goes through vector algebra. He is very thorough. The book cumulates in an introduction to quaternions and tensors. Throughout the book the author raises questions similar to the one above. I found this book very useful for learning both the algebra of vectors and also the ideas behind them. I highly recommend this book for anyone who needs to learn about vector algebra.

The Author points out some critical remarks on the traditional presentation of the notion of "vector". There is the practice to use the same term "vector" for abstract vector (or geometric vector) and for vector quantities. While abstract vectors can be summed, vector quantities can be summed only if they are of the same kind. See the page 14 for this discussion.

Overall, I have to say this is a very well written book. It covers an important topic with very good explanation. However, I don't think it performs a necessary function. I bought this book as a 4th-year engineering student with years of experience with vectors from math, science, and engineering classes. What I found is that this book is a) far too elementary to be useful to anyone with significant experience with vectors, and b) too technical to be useful to anyone with no experience with vectors. Think of an experienced artist reading a book on the use of a paintbrush.It is quite unlikely that a high school math or science teacher would use this book to explain vectors to students. Although the author's explanations are thorough and simple, the notation is too technical for that level. However, if you, like me, thought to buy this book expecting to deepen your understanding of the mathematics of vectors, you will probably be disappointed. This is a nearly complete list of topics of the first 5 of 6 chapters:Definition of vectors, parallelogram law, basic vector algebra (+, -, =), scalar (dot) product, vector (cross) product, and a number of applications of vectors (kinematics, forces & moments, angular motion, etc...)He also introduces tensors in the last chapter, but, in the author's own words: "This being a book about vectors, we have presented only the sketchiest account of tensors ..."If you want a book on vectors for completeness of a personal library, this is a good one to have. But if you just want the information, you probably know 90%+ of what's in this book if you're the kind of person that reads Dover math books or comes from science/engineering.

This is a great basic introduction to vectors. It would be a great supplementary text for undergraduate physics and for AP High School students. The author gives decent explanations about common usages and mathematical techniques involving vectors.I used my copy in the seventies for a quick brush-up prior to taking a grad level physics course. For the price this is a good value.

I hate having to give this book three stars, because I did like it some. I was genuinely torn on whether to give it three or four stars. The problem is that I just didn’t get enough out of the book, and I think only very specific people would get enough out of it. A quick list of people that should read this: a high school or undergraduate college student who wants a quick run-through of vectors before taking a calculus course, anyone earning a Ph.D. (or D.A. or M.A.T., etc.) in mathematics, and philosophers of mathematics. That sounds like an odd listing of people, and it is, which is why I feel the people that will benefit from the book are too few. This is more a consequence of the book’s construction and content than it is the idea of the book.First of all, the book is written to be very conceptually gratifying: the notion of what a vector really is, given the difficulty in defining it and given the often contradicting notions that pop in definitions. The book is geared toward bringing this question to the fore. That’s what makes it nice for philosophers, and I think every Ph.D. in math should have this valuable perspective when teaching. This aspect is only a little bit good for students wanting to learn about vectors. Usually, the way these sorts of conceptual projects work is that you learn something and then you unlearn it by drawing distinction that what was learned cannot handle. That’s philosophy; and it permits greater, more precise definitions and relations in the future.From a pedagogical standpoint, I like the setup of the text, but it doesn’t function well because of the particular layout. The book is expository texts with continuous exercises, causing intermittence in the explanatory material. Some may like this, but I find it horribly counterproductive to have just one concept explained, then jump into questions. Instead, a series of exercises that guide learning prior to the text (like Bogart’s “discovery method”) or a substantial text followed by exercises, which is typical of math texts, would serve the reader much better. I also wonder about how well things were explained in the text. Being competent with tensors, I actually wasn’t sure what Banesh was saying immediately in a couple of instances. This augmented some concerns I had about the earlier bits of text, where I asked myself whether someone reading this for the first time and new to vectors would get the gist. I came to the conclusion that very good readers of mathematics would, but that it would be valuable to less than half of those readers. After all, exceptional readers of mathematics probably don’t need this kind of primer. That makes me think about 20% of all students going into their first calc course would get enough out of this book. Others, still, might find it of value.My general thought on this book is that someone needs to take the idea and do another book that better focuses the thinking of the book. Affine geometry, in my opinion, absolutely needs to be added. I do not fault Hoffmann for not including it, because so few dealing with vectors have found the value in this. However, it seems like this would be a great addition to the book, being that one of his closing comments is a provocative one, pertaining to the fact that he has not introduced a metric tensor (and affine geometry does not entail a measure of length).It’s a good book with a great idea, but I just didn’t get enough out of it.

With the name Hoffmann Banesh, this book need no review. He is the author of many wonderful mathematics books which include illustration. As the old saying "a picture worth a thousand words", this book use picture to solve the mysterious of vector which is so complex. It will help you gain a better understanding of vector if you're taking Physics or Vector Analysis class.

Very good book, and good quality printing.

Easy to read and comprehend topics. If your starting maths and need clear information on vectors, then this is the book for you.