Appendix: Vector Operations
A vector is a quantity which possesses magnitude and direction.
In order to describe a vector mathematically, a coordinate system
having orthogonal axes is usually chosen. In this text, use is made
of the Cartesian, circular cylindrical, and spherical coordinate
systems. In these three-dimensional systems, any vector is completely
described by three scalar quantities. For example, in Cartesian
coordinates, a vector is described with reference to mutually
orthogonal coordinate axes. Then the magnitude and orientation of
the vector are described by specifying the three projections of the
vector onto the three coordinate axes.
In representing a vector A mathematically, its direction
along the three orthogonal coordinate axes must be given. The
direction of each axis is represented by a unit vector i, that
is, a vector of unit magnitude directed along the axis. In Cartesian
coordinates, the three unit vectors are denoted ix, iy,
iz. In cylindrical coordinates, they are ir, i ,
iz, and in spherical coordinates, ir, i ,
i. A, then, has three
vector components, each component corresponding to the projection of
A onto the three axes. Expressed in Cartesian coordinates, a
vector is defined in terms of its components by
These components are shown in Fig. A.1.1.
1 Vectors are usually indicated
either with boldface characters, such as A, or by drawing a
line (or an arrow) above a character to indicate its vector nature,
as in \bar A or \vec A.
Figure A.1.1 Vector A represented by its components in Cartesian coordiantes and unit vectors i.
The sum of two vectors A = Ax ix + Ay iy + Az iz
and B = Bx ix + By iy + Bz iz
is effected by adding the coefficients of each of the components, as
shown in two dimensions in Fig. A.1.2a.
From (2), then, it should be clear that vector addition is both
commutative, A + B = B + A, and associative,
(A + B) + C = A + (B + C).
Figure A.1.2 (a) Graphical representation of vector addition in terms of specific coordinates. (b) Representation of vector addition independent of specific coordinates.
Graphically, vector summation can be performed without regard to the
coordinate system, as shown in Fig. A.1.2b,
by noticing that the sum A + B is a vector directed along
the diagonal of a parallelogram formed by A and B.
It should be noted that the representation of a vector in terms of
its components is dependent on the coordinate system in which it is
carried out. That is, changes of coordinate system will require an
appropriate vector transformation. Further, the variables used must also
be transformed. The transformation of variables and vectors from one
coordinate system to another is illustrated by considering a
transformation from Cartesian to spherical coordinates.
Example 1.2.1. Transformation of Variables and Vectors
We are given variables in terms of x, y, and z and vectors such as
A = Ax ix + Ay iy + Az iz. We
wish to obtain variables in terms of r, , and and
vectors expressed as A = Ar ir + A i +
A i. In Fig. A.1.3a, we see that the point P has two
representations, one involving the variables x, y and z and the
other, r, and . In particular, from
Fig. A.1.3b, x is related to the spherical coordinates by
Figure A.1.3 Specification of a point P in Cartesian and spherical coordinates. (b) Transformation from Cartesian coordinate x to spherical coordinates. (c) Transformation of unit vector in xdirection into spherical coordinates
In a similar way, the variables y and z evaluated in spherical
coordinates can be shown to be
The vector A is transformed by resolving each of the unit
vectors ix, iy, iz in terms of the unit vectors in
spherical coordinates. For example, ix can first be resolved
into components in the orthogonal coordinates (x' ,
y' , z) shown in Fig. A.1.3c. By definition,
y' is along the intersection of the = constant and
the x-y planes. Also in the x - y plane is x', which is
perpendicular to the y' - z plane. Thus, sin , cos
, and 0 are the components of ix along the x' ,
y', and z axes respectively. These components are in turn
resolved into components along the spherical coordinate directions by
recognizing that the component sin along the x' axis
is in the -i direction while the component of cos
along the y' axis resolves into components cos cos
in the direction of i, and cos sin
in the ir direction. Thus,
It must be emphasized that the concept of a vector is independent
of the coordinate system. (In the same sense, in Chaps. 2 and 4,
vector operations are defined independently of the coordinate system
in which they are expressed.) A
vector can be visualized as having the direction and magnitude of an
arrow-tipped line element. This picture makes it possible to deal
with vectors in a geometrical language that is independent of the
choice of a particular coordinate system, one that will now be used to
define the most important vector operations.
For analytical or numerical purposes, the operations are
usually carried out in coordinate notation. Then, as illustrated,
either in the text that follows or in the problems, each operation
will be evaluated in a Cartesian coordinate system.
Definition of Scalar Product
Given vectors A and B as illustrated in Fig. A.1.4, the
scalar, or dot product, between the two vectors is defined as
where is the angle between the two vectors.
Figure A.1.4 Illustration for definition of dot product.
It follows directly from its definition that the scalar product is
The scalar product is also distributive.
To see this, note that A C is the projection of A
onto C times the magnitude of C, |C|, and B
C is the projection of B onto C times
|C|. Because projections are additive, (11) follows.
These two properties can be used to define the scalar product in
terms of the vector components in Cartesian coordinates. According
to the definition of the unit vectors,
With A and B expressed in terms of these components, it
follows from the distributive and commutative properties that
Thus, in agreement with (9), the square of the magnitude of a vector
Definition of Vector Product
The cross-product of vectors A and B is a vector C
having a magnitude
and having a direction perpendicular to both A and B.
Geometrically, the magnitude of C is the area of the
parallelogram formed by the vectors A and B. The vector
C has the direction of advance of a right-hand screw, as though driven
by rotating A into B. Put another way, a right-handed coordinate
system is formed by A - B - C, as is shown in Fig. A.1.5.
The commonly accepted notation for the cross-product is
Figure A.1.5 Illustration for definition of vector-product
It is useful to note that if the vector A is resolved into two
mutually perpendicular vectors, A = A\perp +
A\parallel, where A\perp lies in the plane of A and
B and is
perpendicular to B and A\parallel is parallel to B, then
This equality follows from the fact that both cross-products have equal
magnitude (since |A\perp x B| = |A\perp ||B|
and |A|\perp | = |A| sin ) and direction
(perpendicular to both A and B).
The distributive property for the cross-product,
can be shown using (17) and the geometrical construction in Fig. A.1.6 as
follows. First, note that (A + B)\perp =
(A\perp + B\perp ), where \perp denotes a component in
the planes of A and D or B and D,
respectively, and perpendicular to D. Thus,
Figure A.1.6 Graphical representation showing that the vector-product is distributive.
Now, we need only show that
This equation is given graphical expression in Fig. A.1.6 by the vectors
A\perp , B\perp, and their sum. To within a factor of
|D|, the three vectors A\perp x D,
B\perp x D, and their
sum, are, respectively, the vectors A\perp , B\perp, and
their sum, rotated by 90 degrees. Thus, the vector addition property
already shown for A\perp + B\perp also applies to
A\perp x D + B\perp x D.
Because interchanging the order of two vectors calls for a
reassignment of the direction of the product vector (the direction of
C in Fig. A.1.5), the commutative property does not hold. Rather,
Using the distributive law, the vector product of two vectors can be
constructed in terms of their Cartesian coordinates by using the following
properties of the vector products of the unit vectors.
A useful mnemonic for finding the cross-product in Cartesian
coordinates is realized by noting that the right-hand side of (23)
is the determinant of a matrix:
The Scalar Triple Product
The definition of the scalar triple product of vectors A,
B, and C follows from Fig. A.1.7,
and the definition of the scalar and vector products.
Figure A.1.7 Grpahical representation of scalar triple product.
The scalar triple product is equal to the volume of the parallelepiped
having the three vectors for its three bases. That is, in (25) the
second term in square brackets is the area of the base parallelogram
in Fig. A.1.7 while the first is the height of the parallelopiped.
The scalar triple product is positive if the three vectors form a
right-handed coordinate system in the order in which they are
written; otherwise it is negative. Hence, a cyclic rearrangement in
the order of the vectors leaves the value of the product unchanged.
It follows that the placing of the cross and the dot in a scalar
triple product is arbitrary. The cross and dot can be interchanged
without affecting the product.
Using the rules for evaluating the dot product and the
cross-product in Cartesian coordinates, we have
The Double Cross-Product
Consider the vector product A x (B x C). Is
there another, sometimes more useful, way of expressing this double
cross-product? Since the product B x C is
perpendicular to the
plane defined by B and C, then
the final product A x (B x C) must lie in
the plane of B and C. Hence, the vector product
must be expressible as a linear
combination of the vectors B and C. One way to find the
coefficients of this linear combination is to evaluate the product in
Cartesian coordinates. Here we prefer to use a geometric derivation.
Because the vector B x C is perpendicular to the plane
defined by the vectors B and C, it follows from Fig. A.1.7 that
where A' is the projection of A onto the plane
defined by B and C. Next, we separate the vector C into
a component parallel to B, C\parallel, and a component
perpendicular to B, C\perp, as shown by Fig. A.1.8, so
Figure A.1.8 Graphical representation of double cross-product.
Then, according to the properties of the cross-product, the magnitude
of the vector product is given by
and the direction of the vector product is orthogonal to
A' and lies in the plane defined by the vectors B and
C, as shown in Fig. A.1.8
A rule for constructing a vector perpendicular to a given
vector, A', in an x - y
plane is as follows. First, the two components of A'
with respect to any two orthogonal axes (x, y) are determined.
Here these are the directions of C\perp and B with
components A' C\perp, and A' \cdot
B, respectively. Then, a
new vector is constructed by interchanging the x and y components and
changing the sign of one of them. According to this rule, Fig. A.1.8
shows that the vector A x (B x C) is given by
Now, because C\parallel has the same direction as B,
and addition of (31) gives
Now observe that A' C = A \cdot C
and A' B = A \cdot B (which follow from
the definition of A' as the projection of A into the
B - C plane), and the double cross-product becomes
This result is particularly convenient because it does
not contain any special notation or projections.
The vector identities found in this Appendix are summarized in
Table III at the end of the text.