Hirst A. - Vectors in 2 or 3 Dimensions

This series is designed particularly, but not exclusively, for students reading degree programmes based on semester-long modules. Each text will cover the essential core of an area of mathematics and lay the foundation for further study in that area. Some texts may include more material than can be comfortably covered in a single module, the intention there being that the topics to be studied can be selected to meet the needs of the student. Historical contexts, real life situations, and linkages with other areas of mathematics and more advanced topics are included. Traditional worked examples and exercises are augmented by more open-ended exercises and tutorial problems suitable for group work or self-study. Where appropriate, the use of computer packages is encouraged. The first level texts assume only the A-level core curriculum.

Professor Chris D. Collinson, , DrJohnston Anderson and Mr. Peter Holmes

Now that vectors are no longer in the core of all A-level mathematics syllabuses, the first time some students meet vectors will be in the first year at university. It seemed a good idea, therefore, to have a book on vectors starting at a very basic level, but also incorporating many of the ideas found in university mathematics courses. This book is written from the geometrical standpoint (indeed, the title was originally to have been Geometric vectors), and although applications to mechanics will be pointed out from time to time, and techniques from linear algebra will be employed, it is the geometric view which will be emphasised throughout.

, a project topic is suggested which would be well within the scope of a good first-year undergraduate (or even someone studying further mathematics at A-level).

The first four chapters are intended to be an introduction to the properties of vectors and the techniques for using them to find vector equations of planes, lines and intersections of planes and planes, planes and lines, and lines and lines. These vector methods are also the easiest way of finding the cartesian equations of planes and lines in three dimensions.

. Not many years ago, transformation geometry in two dimensions would have been met at GCSE level (or O-level, as it was then). Now this is no longer the case, it seems appropriate to include this topic in a book on geometric vectors, as, when one fully appreciates the two-dimensional case, the progression to three-dimensional transformation geometry is a small step. The basic geometrical results are expanded upon and linked with eigenvalues and eigenvectors, looked at again in a geometric light, and some useful special cases are considered.

world of manifolds and tangent bundles. These last three chapters are both a vehicle for consolidating the ideas found earlier in the book and a stepping stone to more advanced theories.

I would like to thank my colleague Ray dInverno for suggesting that I write this book, Professor Chris Collinson, who, as editor of the series, encouraged me from afar, and my colleagues and students who have given me so many new slants on the subject matter over the years. Thanks go to David Firth for introducing me to the delights of PICTEX for preparing diagrams, and to many of my colleagues whose brains have been picked concerning the use of TEX in which this text was prepared. I am particularly grateful to Susan Ward who read through the chapters of this book as they were written. Her careful attention to detail has helped to eradicate many of the host of errors that occurred in a first draft, and her view as a student on the receiving end was invaluable. That she found time to do this while studying full time for a degree and looking after her family is a tribute to her ability to organise her time so competently. Lastly, and most important of all, my thanks go to my husband, Keith, without whose constant encouragement and support (both moral and gastronomic) this book would not have been written.