Vectors and Lines in Space

In the preceding section, we used the algebra of vectors to prove some facts from plane geometry. This approach really comes into its own in solid geometry. Without using vectors, it is quite hard to define such basic concepts as a straight line, or the angle between two lines, in space. In this section we shall develop geometry in three-dimensional space with vectors as our starting point. The notions of a straight line and an angle in space will be defined using vectors, and we shall use vector algebra to solve problems about lines and angles. Later on in this chapter we shall continue our development of solid geometry, using vectors to study planes in space.

Figure 10.3.1

Vectors in space are developed in the same way as vectors in the plane. Three-dimensional space has three perpendicular coordinate axes, x, y, and z, as shown in Figure 10.3.1. This is called a right-handed coordinate system, because the right thumb, forefinger, and middle finger can point in the direction of the positive x, y, and z axes respectively.