4 Euclidean Geometry
several variables. Vector algebra and vector analysis were developed in the
1880s, independently by Gibbs and by a self-taught British engineer OLIVER
HEAVISIDE (1850-1925). In their developments, both Gibbs and Heaviside
were motivated by applications to physics: many physical quantities, such
as position, velocity, acceleration, and force, can be represented by vectors.
All constructions in vector algebra and analysis are not tied to any
particular coordinate system in M^3 , and do not rely on the interpretation of
vectors as position vectors. Nevertheless, it is convenient to depict a vector
as a line segment with an arrowhead at one end to indicate direction, and
think of the length of the segment as the magnitude of the vector.
Remark 1.1 Most of the time, we will identify all the vectors having
the same direction and length, no matter the starting point. Each vec-
tor becomes a representative of an equivalence class of vectors and can be
moved around by parallel translation. While this identification is convenient
to study abstract properties of vectors, it is not always possible in certain
physical problems (Figure 1.1.1).
F\ F2 F2 Fi
-« 1 I *- *i N
Stretching Compressing
Fig. 1.1.1 Starting Point of a Vector Can Be Important!1.1.3 Euclidean Space as a Linear Space
Consider the Euclidean space K^3 and choose a point O to serve as the
origin. In mechanics this is sometimes referred to as choosing a frame of
reference, or frame for short. As was mentioned in Remark 1.1, we assume
that all the vectors can be moved to the same starting point; this starting
point defines the frame. Accordingly, in what follows, the word frame will
have one of the three meanings:
- A fixed point;
- A fixed point with a fixed coordinate system (not necessarily Carte-
sian) ; - A fixed point and a vector bundle, that is, the collection of all
vectors that start at that point.