Chapter 3 - Vectors and Analytic Geometry

Abstract

The only subspaces of R3, other than the zero subspace and R3 itself, are those of dimensions one and two. By definition, a one-dimensional subspace consists of all multiples of a fixed nonzero vector and a two-dimensional subspace consists of all linear combinations of two linearly independent vectors. These subspaces have familiar geometric interpretations. The chapter describes lines and planes. It explains coplanar and collinear vectors. Three vectors are coplanar if they are linearly dependent and noncoplanar if they are not linearly dependent. Two vectors U and V are linearly dependent if and only if line segments drawn from the origin in the directions of U and V are on the same line; hence, two vectors are said to be collinear if they are parallel and noncollinear if they are not parallel. The chapter discusses the use of vectors to study the motion of a particle in R3. It considers position, velocity, and acceleration vectors. The chapter also explores parametric functions, that is, vector valued functions of a single real variable. It describes two coordinate systems in R3, namely, cylindrical coordinates and spherical coordinates.