文档介绍：Chapter Two
Vectors-Algebra and Geometry
Vectors
A directed line segment in space is a line segment together with a direction. Thus
the directed line segment from the point P to the point Q is different from the directed
line segment from Q to P. We frequently denote the direction of a segment by drawing an
arrow head on it pointing in its direction and thus think of a directed segment as a spear.
We say that two segments have the same direction if they are parallel and their directions
are the same:
Here the segments L1 and L2 have the same direction. We define two directed segments L
and M to be equivalent ( L ***@M ) if they have the same direction and have the same
length. An equivalence class containing a segment L is the set of all directed segments
equivalent with L. Convince yourself every segment in an equivalence class is equivalent
with every other segment in that class, and two different equivalence classes must be
disjoint. These equivalence classes of directed line segments are called vectors. The
members of a vector v are called representatives of v. Given a directed segment u, the
vector which contains u is called the vector determined by u. The length, or magnitude,
of a vector v is defined to be mon length of the representatives of v. It is generally
designated by |v|. The angle between two vectors u and v is simply the angle between the
directions of representatives of u and v.
Vectors are just the right mathematical objects to describe certain concepts in
physics. Velocity provides a ready example. Saying the car is traveling 50 miles/hour
doesn’t tell the whole story; you must specify in what direction the car is moving. Thus
velocity is a vector-it has both magnitude and direction. Such physical concepts abound:
force, displacement, acceleration, etc. The real numbers (or sometimes, plex
numbers) are frequently called scalars in order to distinguish them from vectors.
We now introduce an arithmetic, or algebr