2 Vectors and geometry
in
three dimensions
2.1 Vectors in E3 To facilitate discussions and solutions of problems
in geometry and calculus, and for many other purposes in pure and
applied mathematics, it is necessary to know about things called vectors.
All points and vectors with which we are concerned are supposed to lie in
ordinary Euclid space E3 of three dimensions in which such things as
points, lines, planes, cubes, spheres, and automobiles can exist. The
definitions of this section do not depend upon a coordinate system and are
therefore said to be intrinsic definitions. We shall hear more about this
matter later.
Before introducing vectors, we observe the familiar fact that two
distinct (that is, different) points P, and P2 determine the line P1P2 which
passes through P1 and P2 and extends beyond P1 and P2 in two directions
as in Figure 2.11. Vectors are more like line segments than like lines.
An ordered pairP1,P2 of distinct points, in which P1 and P2 are respectively
the first point and the second point in the pair, determines the vector PiP2
or "arrow" or "directed line segment" which runs (or extends) from the
48