文档介绍：Chapter One
Euclidean Three-Space
Introduction.
Let us briefly review the way in which we established a correspondence between
the real numbers and the points on a line, and between ordered pairs of real numbers and
the points in a plane. First, the line. We choose a point on a line and call it the origin.
We choose one direction from the origin and call it the positive direction. The opposite
direction, not surprisingly, is called the negative direction. In a picture, we generally
indicate the positive direction with an arrow or a plus sign:
Now we associate with each real number r a point on the line. First choose some
unit of measurement on the line. For r > 0, associate with r the point on the line that is a
distance r units from the origin in the positive direction. For r < 0, associate with r the
point on the line that is a distance r units from the origin in the negative direction. The
number 0 is associated with the origin. A moments reflection should convince you that
this procedure establishes a so-called one-to-one correspondence between the real
numbers and the points on a line. In other words, a real number determines exactly one
point on a line, and, conversely, a point on the line determines exactly one real number.
This line is called a real line.
Next we establish a one-to-one correspondence between ordered pairs of real
numbers and points in a plane. Take a real line, called the first axis, and construct another
real line, called the second axis, perpendicular to it and passing through the origin of the
first axis. Choose this point as the origin for the second axis. Now suppose we have an
ordered pair (x1 , x2 ) of reals. The point in the plane associated with this ordered pair is
found by constructing a line parallel to the second axis through the point on the first axis
corresponding to the real number x1 , and constructing a line parallel to the first axis
through the point on the second axis corresponding