Chapter 1: Vectors, Lines, and Quadric Surfaces

Single-variable calculus lives on a line and draws its pictures in a plane. Everything in this book happens in space, and that one extra dimension changes what a picture is: a flat screen showing a three-dimensional scene hides exactly one dimension of information, and it will lie to you cheerfully unless you move your point of view. That is why every figure in this chapter is one you can grab and orbit. Your first job is not computation — it is learning to distrust a single viewpoint and to resolve depth by looking again from somewhere else.

We start with the geography of space itself: coordinates, the eight octants, and the distance formula, whose most famous consequence is the sphere. Then come lines, which hold a genuine surprise — in space, two lines can fail to meet without being parallel, a situation impossible in the plane. Deciding whether lines cross, run parallel, or pass by each other (the skew case) turns out to be a question about solving three equations with two unknowns.

Finally we meet the quadric surfaces, the three-dimensional cousins of the conic sections. Rather than memorizing a zoo of shapes, you will learn one skill that classifies all of them: slicing. Hold a coordinate fixed, look at the curve the slice produces — a trace — and let the family of traces tell you the anatomy of the surface. The chapter closes with the hyperbolic paraboloid, a saddle whose traces flip character as the slicing plane sweeps through one special height.