12 - Analytic Geometry: Part 2 — Geometric Representation of Vectors and Algebraic Operations

Abstract

This chapter continues with the pre-chemometrics review of analytic geometry, noting the term “vector” in all cases can be represented by a matrix of r x c dimensions, where r = number of rows and c = number of columns. The operations defined are employed in future discussions. If M represents a vector with components (or elements) as (Mx, My), then sM (where s is a real number, also termed a “scalar”) is defined as the vector represented by (sMx, sMy); and the length of sM is s times the length of M. In the case when s < 0 (s is a positive, real number), the vectors sM and M have the exact same direction. For the case where s < 0 (where s is a negative, real number), then the vectors sM and M have the exact opposite directions. When s = 0, there is no definition for the vector or direction. Thus, the direction angles of M can be related to those of sM. Vector division is represented as vector multiplication by using a fractional multiplier term. This chapter presents examples of vector-vector addition and vector-vector subtraction along with their graphical representations.