associate each base vector with a particular coordinate direction. Most vector quantities used in
this book, such as the position vector or velocity
vector, are free to take on any orientation and
therefore do not carry subscripts. The coordinate
axes, and therefore the base vectors, comprise a
right-handed system. By right handed we mean
that the axes (Ox, Oy, Oz) extend, respectively, in
the same mutual orientations as the thumb,
index finger, and middle finger of the right hand
when these fingers are directed orthogonal to one
another. Base vectors need not be orthogonal to
one another nor of unit magnitude, and Cartesian
coordinates can be left-handed, but these alternatives are not employed in this book.
The scalar components of a vector are quantities that can be visualized as the orthogonal projections of the vector arrow onto lines in arbitrary
but designated directions. In general, scalar components are written with the same symbol as the
vector, but the type is not bold and each component has a subscript that identifies the line projected upon. For the position vector, p, the
designated directions are parallel to the base
vectors (Fig. 2.6b), in other words lines parallel to
the respective coordinate axes. The three components of the position vector are written (p x , p y , p z ).
Thus, the equation for the position vector with
the basis (e x , e y , e z ) is written as the following
linear combination of scalar components and
base vectors:
(2.5)
Although it is convenient to speak of a position
vector simply as p, and this notation is useful for
writing compact vector equations, most calculations are done using the scalar components.
Apparently Descartes introduced the notion of
using vector components, defined with respect to
a coordinate system in order to develop analytical
geometry (Fung, 1969, p. 22).
An arbitrary vector, v, may have its tail at any
point (x 1 , y 1 , z 1 ) and its head at any other point (x 2 ,
y 2 , z 2 ) so this vector is written:
(2.6)
The quantities (v x , v y , v z ) are the scalar components
of the vector, which can be interpreted as the
ϭ v x e x ϩ v y e y ϩ v z e z
v ϭ (x 2 Ϫ x 1 )e x ϩ ( y 2 Ϫ y 1 )e y ϩ (z 2 Ϫ z 1 )e z
p ϭ p x e x ϩ p y e y ϩ p z e z
orthogonal projections of v onto lines drawn
parallel to the base vectors (e x , e y , e z ), respectively.
If the two points are coincident v ϭ 0 ϭ (0, 0, 0) is
the zero vector.
The meter (m) is the physical unit used to
measure distance, and this is carried by the position vector components, not the base vectors.
Thus one would write, for example, p y ϭ 22.5 m
and |e y | ϭ 1. Each base vector has a magnitude of
one but does not carry a physical unit. The base
vectors provide the directional information necessary to compose a position vector. The threedimensional space defined in terms of position
vector components using the basis (e x , e y , e z ) is
referred to as Euclidean space (Lipschutz, 1969).
When the base vectors change direction
because of a rotation of the coordinate system
about the origin (Fig. 2.6c), the components of the
position vector p for an arbitrary point P change,
but the magnitude and direction of the position
vector do not change. For example, the projection
p z Ј of p onto the rotated coordinate axis OzЈ is different from the projection p z onto the original
coordinate axis Oz. Thus we say that (p x , p y , p z ) are
the components of p with respect to a particular
basis (e x , e y , e z ), and this basis must be defined in
order to interpret the components. On the other
hand if the base vectors and coordinate system are
translated to a new origin, OЈ, but not rotated (Fig.
2.6d), a new position vector, pЈ, extends to the
given point, P, and this vector has a different magnitude and/or direction than p. In general, two
vectors are equal only if they have the same magnitude and direction, so the position vectors for
the point P referred to different origins are not
equal, despite the fact that they define the location of the same point. In this sense position
vectors are so-called fixed vectors (Malvern, 1969):
they emanate from a particular point, the origin
of the coordinate system, and the location of this
origin must be specified in order to define a set of
position vectors. Position vectors are useful quantities for defining the locations of points, but they
lack the attributes of those vectors that we think
of as physical entities, such as force or velocity,
which are independent of an arbitrarily defined
coordinate system.
The magnitude of any vector is calculated as
the square root of the sum of the squares of its
2.2 LOCAL COORDINATES AND POSITION VECTORS
37
this book, such as the position vector or velocity
vector, are free to take on any orientation and
therefore do not carry subscripts. The coordinate
axes, and therefore the base vectors, comprise a
right-handed system. By right handed we mean
that the axes (Ox, Oy, Oz) extend, respectively, in
the same mutual orientations as the thumb,
index finger, and middle finger of the right hand
when these fingers are directed orthogonal to one
another. Base vectors need not be orthogonal to
one another nor of unit magnitude, and Cartesian
coordinates can be left-handed, but these alternatives are not employed in this book.
The scalar components of a vector are quantities that can be visualized as the orthogonal projections of the vector arrow onto lines in arbitrary
but designated directions. In general, scalar components are written with the same symbol as the
vector, but the type is not bold and each component has a subscript that identifies the line projected upon. For the position vector, p, the
designated directions are parallel to the base
vectors (Fig. 2.6b), in other words lines parallel to
the respective coordinate axes. The three components of the position vector are written (p x , p y , p z ).
Thus, the equation for the position vector with
the basis (e x , e y , e z ) is written as the following
linear combination of scalar components and
base vectors:
(2.5)
Although it is convenient to speak of a position
vector simply as p, and this notation is useful for
writing compact vector equations, most calculations are done using the scalar components.
Apparently Descartes introduced the notion of
using vector components, defined with respect to
a coordinate system in order to develop analytical
geometry (Fung, 1969, p. 22).
An arbitrary vector, v, may have its tail at any
point (x 1 , y 1 , z 1 ) and its head at any other point (x 2 ,
y 2 , z 2 ) so this vector is written:
(2.6)
The quantities (v x , v y , v z ) are the scalar components
of the vector, which can be interpreted as the
ϭ v x e x ϩ v y e y ϩ v z e z
v ϭ (x 2 Ϫ x 1 )e x ϩ ( y 2 Ϫ y 1 )e y ϩ (z 2 Ϫ z 1 )e z
p ϭ p x e x ϩ p y e y ϩ p z e z
orthogonal projections of v onto lines drawn
parallel to the base vectors (e x , e y , e z ), respectively.
If the two points are coincident v ϭ 0 ϭ (0, 0, 0) is
the zero vector.
The meter (m) is the physical unit used to
measure distance, and this is carried by the position vector components, not the base vectors.
Thus one would write, for example, p y ϭ 22.5 m
and |e y | ϭ 1. Each base vector has a magnitude of
one but does not carry a physical unit. The base
vectors provide the directional information necessary to compose a position vector. The threedimensional space defined in terms of position
vector components using the basis (e x , e y , e z ) is
referred to as Euclidean space (Lipschutz, 1969).
When the base vectors change direction
because of a rotation of the coordinate system
about the origin (Fig. 2.6c), the components of the
position vector p for an arbitrary point P change,
but the magnitude and direction of the position
vector do not change. For example, the projection
p z Ј of p onto the rotated coordinate axis OzЈ is different from the projection p z onto the original
coordinate axis Oz. Thus we say that (p x , p y , p z ) are
the components of p with respect to a particular
basis (e x , e y , e z ), and this basis must be defined in
order to interpret the components. On the other
hand if the base vectors and coordinate system are
translated to a new origin, OЈ, but not rotated (Fig.
2.6d), a new position vector, pЈ, extends to the
given point, P, and this vector has a different magnitude and/or direction than p. In general, two
vectors are equal only if they have the same magnitude and direction, so the position vectors for
the point P referred to different origins are not
equal, despite the fact that they define the location of the same point. In this sense position
vectors are so-called fixed vectors (Malvern, 1969):
they emanate from a particular point, the origin
of the coordinate system, and the location of this
origin must be specified in order to define a set of
position vectors. Position vectors are useful quantities for defining the locations of points, but they
lack the attributes of those vectors that we think
of as physical entities, such as force or velocity,
which are independent of an arbitrarily defined
coordinate system.
The magnitude of any vector is calculated as
the square root of the sum of the squares of its
2.2 LOCAL COORDINATES AND POSITION VECTORS
37