SEVENTEEN - VECTORS IN SPACE

Abstract

This chapter discusses the development of a theory of vectors in the plane to the vectors in space and discusses that there are great similarities between the two notions. Any point in space can be represented by an ordered triple of real numbers. (a, b, c), where a, b, and c are real numbers. The set of ordered triples of that form is called real three-dimensional space and is denoted R3. It is the most common representation of a point in R3, which is very similar to the representation of a point in the plane by its x- and y-coordinates; however, there are many ways to represent a point in R3. The chapter discusses the vectors in R3 and lines in R3 and describes the properties of vector functions in R3. It describes a new product called the cross product or vector product, which is defined only in R3. A surface in space is defined as the set of points satisfying the equation F(x, y, z) = 0. For example, the equation F(x, y, z) = x2 + y2 + z2 − 1 = 0 is the equation of the unit sphere. The chapter discusses quadric surfaces. It also reviews some of the most commonly encountered surfaces in R3. The chapter discusses two common ways to represent points in space. The first is the generalization of the polar coordinate system in the plane. The second new coordinate system in space is the spherical coordinate system.