Relative Position and Position Vectors 31.1.2 Relative Position and Position Vectors
Our intuitive conception and observation of position and motion suggest
that the position of a point in space can only be specified relative to some
other point, chosen as a reference. Likewise, the motion of a point can only
be specified relative to some reference point.
The view that only relative motion exists and no meaning can be given
to absolute position or absolute motion has been advocated by many promi-
nent philosophers for many centuries. Among the famous proponents of this
relativistic view were the Irish bishop and philosopher GEORGE BERKELEY
(c. 1685-1753), and the Austrian physicist and philosopher ERNST MACH
(1836-1916). An opposing view of absolute motion also had prominent
supporters, such as Sir ISAAC NEWTON (1642-1727). In 1905, the German
physicist ALBERT EINSTEIN (1879-1955) and his theory of special relativ-
ity seemed to resolve the dispute in favor of the relativists (see Section 2.
below).
Let us apply the idea of the relative position to points in the Euclidean
space M^3. We choose an arbitrary point O as a reference point and call it an
origin. Relative to O, the position of every point P in R^3 is specified by the
directed line segment r = OP from O to P. This line segment has length
||r|| — \OP\, the distance from O to P, and is called the position vector
of P relative to O (the Latin word vector means "carrier"). Conversely, any
directed line segment starting at O determines a point P. This description
does not require a coordinate system to locate P.
In what follows, we denote vectors by bold letters, either lower or
upper case: u, R. Sometimes, when the starting point O and the
ending point P of the vector must be emphasized, we write OP to
denote the corresponding vector.
The position vectors, or simply vectors, can be added and multiplied
by real numbers. With these operations of addition and multiplication, the
set of all vectors becomes a vector space. Because of the special geometric
structure of K^3 , two more operations on vectors can be defined, the dot
product and the cross product, and this was first done in the 1880s by the
American scientist JosiAH WlLLARD GlBBS (1839-1903). We will refer to
the study of the four operations on vectors (addition, multiplication by real
numbers, dot product, cross product) as vector algebra. By contrast,
vector analysis (also known as vector calculus) is the calculus on M^3 ,
that is, differentiation and integration of vector-valued functions of one or