Chapter 12 Vectors and Geometry of Space 12.1 Three-Dimensional Coordinate Systems *12.2 Vectors *12.3 The Dot Product *12.4 The Cross Product 12.5 Equations of Lines and Planes 12.6 Cylinders and Quadric Surfaces *12.7 Cylindrical and Spherical Coordinates

In this chapter we introduce vectors and coordinate systems for three-dimensional space. This will be the setting for our study of the calculus of functions of two variables in Chapter 14 because the graph of such a function is a surface in space. In this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space.

12.1 Three-Dimensional Coordinate Systems Coordinate axes xz-plane Coordinate planes

Three-Dimensional Rectangular Coordinate Systems Through point O , three axes vertical each other, by right-hand rule, we obtain a Three-Dimensional Rectangular Coordinate Systems Ⅱ origin O Ⅲ axes Ⅳ Ⅰ planes xz-plane octants Ⅶ Ⅵ Ⅴ Ⅷ

The Cartesian product is the set of all ordered triples of real numbers and is denoted by . We have given a one-to-one correspondence between points P in space and ordered triples (a,b,c) in

In three-dimensional analytic geometry , an equation in x, y, and z represents a surface in . For example:

Example

Example

Distance Formula in Three Dimensions The distance between the points and is

Equation of a Sphere An equation of a sphere with center (h,k,l) and radius r is . In particular, if the center is the origin O, then an equation of the sphere is

12.5 Equations of Lines and Planes Parametric equations of lines Symmetric equations of lines General equations of planes

12.6 Cylinders and Quadric Surfaces A cylinder is a surface that consists of all lines that are parallel to a given line and pass through a given plane curve.

Quadric Surfaces A quadric surface is the graph of a second- degree equation in three variables x, y, and z. The most general such equation is ellipsoid

Elliptic paraboloid

Hyperbolic paraboloid