The Basic Concepts of Vectors

A vector in geometry is both an arrow and an ordered list of numbers — and these two views have to be mastered together: the first supplies the intuition, the second supports the computation. This chapter establishes the foundations of that dual viewpoint.

Starting from free vectors and unit vectors, it explains why a vector can be translated without changing its essence; then it defines the magnitude and the direction, and explains why “two vectors are equal” means that both the magnitude and the direction agree rather than that the endpoints coincide. The criteria for collinearity and coplanarity give the most elementary characterisation of linear relations, and are the starting point of every geometric problem that follows.

The chapter also introduces the coordinate representation of a vector in space, which is the crucial step that turns a geometric problem into an algebraic one — once you have coordinates, geometric propositions such as collinearity and coplanarity can all be verified with determinants or equations.