1. 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

2. 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.

3. 12.1 Three-Dimensional Coordinate Systems
Coordinate axes
xz-plane
Coordinate planes

4. 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

5. 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

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

7. Example

8. Example

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

10. 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

11. 12.5 Equations of Lines and Planes
Parametric equations of lines
Symmetric equations of lines
General equations of planes

12. 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.

13. 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

14. Elliptic paraboloid

15. Hyperbolic paraboloid