CHAPTER 3 - Vector and Matrix Concepts from a Geometric Viewpoint

Abstract

This chapter discusses vector and matrix concepts from a geometric viewpoint. The length of a vector's projection is given by the vector's coordinate on the x, y, and z axes, respectively, and the sign of its projection on x, y, and z depends on where the projection terminates, relative to the origin. A Euclidean space of n dimensions is the collection of all n-component vectors for which the operations of vector addition and multiplication by a scalar are permissible. For any two vectors in the space, there is a nonnegative number, called the Euclidean distance between the two vectors. By appropriate parallel displacement to a position vector, any vector in the space can be represented by its direction angles and length. Any position vector is uniquely determined by knowledge of its magnitude and direction. For any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of their included angle θ.