16 Vectors
16.1 Concepts
Physical quantities such as mass, temperature, and distance have values that are
specified by single real numbers, in appropriate units; for example, 3 1 kg, 273 1 K, and
121 m. Such quantities, having magnitudeonly, are called scalarquantities and obey
the rules of the algebra of real numbers. Other physical quantities, called vectors,
require both magnitude and directionfor their specification. For example, velocity
is speed in a given direction, the speed being the magnitude of the velocity vector.
Other examples are force, electric field, magnetic field, and displacement. To qualify
as vectors these quantities must obey the rules of vector algebra.
1Vector notation and
vector algebra are important for the formulation and solution of physical problems
in three dimensions; in mechanics, fluid dynamics, electromagnetic theory, and
engineering design. Some of these uses of vectors, important in molecular dynamics,
spectroscopy, and theoretical chemistry, are discussed in examples throughout this
chapter.
A vector is represented graphically by a directed line segment; that is, a segment of
line whose length is the magnitude of the vector, in appropriate units, and whose
orientation in space, together with an arrowhead, gives the direction.
Figure 16.1 shows two graphical representations of the same vector, and two ways of
representing it in print. In (a), the vector ais represented by an arrow; in (b), A and B
are the initialand terminalpoints of the vector, and the notation is particularly
useful when the vector is a displacement in space. The length, magnitude, or modulus
of ais written as|a|or simply as a(a scalar). A vector of unit length is called a unit
vector. A vector of zero length is call the null vector 0 , and no direction is defined in
this case.
AB
1Vector algebra has its origins in the algebra of quaternions discovered by Hamilton in 1843 and in
Grassmann’s theory of n-dimensional vector spaces of 1844. The modern notation for vectors in three dimensions
is due to Gibbs.
William Rowan Hamilton (1805 –1865), born in Dublin, is reputed to have been fluent in Latin, Greek, the
modern European languages, Hebrew, Persian, Arabic, Sanskrit, and others at the age of ten. He entered Trinity
College, Dublin, in 1823 and became Astronomer Royal of Ireland and Professor of Astronomy in 1827 without
taking his degree. He is best known for the reformulation and generalization of the mechanics of Newton, Euler,
and Lagrange that became important in the formulation of statistical and quantum mechanics.
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a
a=
−→
ab
a
b
(a) (b)
Figure 16.1