This is a textbook on vectors at the undergraduate/advanced undergraduate level. Its target readership is the undergraduate student of science and engineering. It may also be used by professional scientists and engineers to brush up on various aspects of vectors and applications of their interest. Vectors, vector operators and vector analysis form the essential background to and the skeleton of many courses in science and engineering. Therefore, the utility of a book which clearly builds up the theoretical structure and applications of vectors cannot be over-emphasized. The present book is an attempt to fulfill such a requirement. This book, for instance, can be used to give a course forming a common pre-requisite for a number of science and engineering courses. In this book, I have tried to develop the theory and applications of vectors from scratch. Although the subject is presented in a general setting, it is developed in 3-D space using basic vector algebra. A coordinate-free approach is taken throughout, so that all developments are free of any particular coordinate system and apply to all coordinate systems. This approach directly deals with vectors instead of their components or coordinates and combines these vectors using vector algebra.

A large part of this book is inspired by the geometric algebra of multivectors that originated in the 19th century, in the works of Grassmann and Clifford and which has had a powerful re-incarnation with enhanced applicability in the recent works of D. Hestenes and others [7, 10, 11]. This is one of the most general algebraic formulations of geometry of which vectors form a special case. Keeping the multivector geometric algebra at the backdrop makes the coordinate free approach for vectors emerge naturally. On a personal note, the book on classical mechanics by D. Hestenes [10], which introduced me to the multivector geometric algebra, has always been a source of joy and education for me. I have always enjoyed solving problems from this book, many of them are included here. In fact I have used Hestenes’ work in various places throughout the book, without using or referring to the geometric algebra or geometric calculus.

While designing this book I was guided by two principles: A consistent development of the subject from scratch, and also showing the beauty of the whole edifice and extending the utility of the book to the largest possible cross-section of students.