Chapter 10 Basic Concepts of Descriptive Vector Geometry

Section 10.1 From Arrows to Vectors

10.1.1 Introduction

The basic idea underlying the mathematical concept of a vector is rooted in physics. In science there are quantities that are described by one single number, their magnitude. Such quantities include voltage, work, or power. In mathematical terms, these quantities are simply described by elements of the set of real numbers. There are other quantities which have not only a certain magnitude but also a certain direction, such as force or velocity. For example, a force that acts on a body at a certain point can be visualised as an arrow of a corresponding length starting at that point. The direction of the arrow then corresponds to the direction in which the force acts. This is illustrated in the figure below.

In mathematics, such quantities are described by vectors. In the sciences, the magnitudes of vectors have certain units of measure (e.g. forces are measured in Newton). From a purely mathematical point of view, however, these units of measure are not relevant, so they are omitted here. The concept of a vector is the main topic of this section and will be discussed in detail in Subsection 10.1.3. Since vectors are considered not only in two-dimensional space (i.e. in the plane) but also in three-dimensional space (i.e. in the space), the concept of coordinate systems and points introduced in Module 9 will be extended to three dimensions. This is done in Subsection 10.1.2. Finally, we see that certain operations can be applied to vectors. In Subsection 10.1.4 we will study how these vector operations are carried out.