Page 159 - Schaum's Outline of Theory and Problems of Advanced Calculus

Vectors

VECTORS
The foundational ideas for vector analysis were formed indepen-
Q
dently in the nineteenth century by William Rowen Hamilton and
Herman Grassmann. We are indebted to the physicist John Willard
Gibbs, who formulated the classical presentation of the Hamilton A or A
viewpoint in his Yale lectures, and his student E. B. Wilson, who B
considered the mathematical material presented in class worthy of
organizing as a book (published in 1901). Hamilton was searching for P
a mathematical language appropriate to a comprehensive exposition
of the physical knowledge of the day. His geometric presentation Fig. 7-1
emphasizing magnitude and direction, and compact notation for the
entities of the calculus, was refined in the following years to the benefit of expressing Newtonian
mechanics, electromagnetic theory, and so on. Grassmann developed an algebraic and more philo-
sophic mathematical structure which was not appreciated until it was needed for Riemanian (non-
Euclidean) geometry and the special and general theories of relativity.
Our introduction to vectors is geometric. We conceive of a vector as a directed line segment PQ
ƒ!
from one point P called the initial point to another point Q called the terminal point.We denote vectors
~
ƒ!
A
by boldfaced letters or letters with an arrow over them. Thus PQ is denoted by A or A as in Fig. 7-1.
~
ƒ!
A
The magnitude or length of the vector is then denoted by jPQj, PQ, jAj or jAj.
Vectors are defined to satisfy the following geometric properties.
GEOMETRIC PROPERTIES
1. Two vectors A and B are equal if they have the same magnitude and direction regardless of their
initial points. Thus A ¼ B in Fig. 7-1 above.
In other words, a vector is geometrically represented by any one of a class of commonly
directed line segments of equal magnitude. Since any one of the class of line segments may be
chosen to represent it, the vector is said to be free.In certain circumstances (tangent vectors,
forces bound to a point), the initial point is fixed, then the vector is bound. Unless specifically
stated, the vectors in this discussion are free vectors.
2. A vector having direction opposite to that of vector A but with the same magnitude is denoted
by A [see Fig. 7-2].
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