Synopsis "Vector Algebra: for Engineers and Scientists"
The concept of the vector plays an extremely important role in Engineering, Physics and Geometry. Vector quantities have both magnitude and direction, as opposed to scalar quantities which have only magnitude. For example, the velocity, the acceleration, the force, the electric and magnetic fields, etc. are vector quantities, while mass, temperature, volume, etc. are scalar quantities.Vectors are important in almost all branches of Engineering, Geometry and Physics and in particular in the study of Applied Mathematics. Using vectors, many important equations in Engineering and Physics are expressed in a compact and concise form, independent from the particular coordinate system being used. In this book we lay out fundamental concepts and definitions, define the fundamental vector operations (equality of vectors, addition, subtraction, multiplication of a vector by a scalar, etc), define the various types of vector products (the dot or scalar product, the cross or outer product, the scalar triple product and the vector triple product), and show the strength of vector algebra in proving various important formulas in Geometry, Trigonometry, Engineering and Physics. The book contains 11 chapters, as shown analytically in the Table of contents. The first two chapters are devoted to fundamental concepts, definitions, terminology and vector operations. Chapter 3 is devoted to the Cartesian systems and the coordinate expression of vectors. In chapter 4 we introduce the concept of linear independence of vectors and investigate a number of useful consequences. Chapters 5 up to 9 are devoted to the study of various types of vector products, i.e. the dot product, the cross product, the scalar triple product and the vector triple product, and investigate a considerable number of applications in Physics and Geometry. In chapter 10 we derive the vector equations of straight lines, planes, circles and spheres and prove various properties using the theory of vectors. Finally, in chapter 11 we derive and summarize some fundamental formulas of plane and solid analytic Geometry, (distance of a point from a straight line, distance of a point from a plane, the least distance between two skew lines, the area of a triangle, the volume of a parallelepiped formed by three concurrent vectors, the angle between two planes, etc).The book contains 72 illustrative worked out examples and 145 graded problems for solution. The examples and the problems are designed to help students to develop a solid background in the algebra of vectors, to broaden their knowledge and sharpen their analytical skills and finally to prepare them to pursue successfully more advanced studies in Engineering and Mathematics.
