Analytic Geometry of Space

Once you have coordinates, the points, lines and planes of space can be described by equations. The distance formulas and the angle formulas are then the standard tools for translating a “geometric relationship” into an “algebraic condition”.

This chapter starts from the coordinates of a point in space, builds up the equations of planes and of lines in turn, and then discusses the positional relationships between them: parallel, intersecting, perpendicular. Each relationship comes with several ways of deciding it — through the direction vectors and the normal vectors, through the nature of the solutions of a system of equations, and through the distance formulas — and each has its own scenario of application.

The distance formulas and the angle formulas are the landing point of this chapter: the former give the distance from a point to a plane and from a point to a line, and the latter give the angle between two lines and between two planes. With them in hand you have the tools you need later on, when finding the tangent planes of quadric surfaces and discussing the positional relationships between surfaces.