Vectors will most frequently be denoted throughout the text by
a vector arrow over the symbol representing the quantity in
question. Examples are velocity 
and force 
.
We can represent any vector 
symbolically in therms
of the vector addition of three mutually perpendicular
vectors as follows:
where 
, 
and 
are unit vectors in the
x, y, z directions, respectively, and 
, 
and 
are the scalar components of the vector for the
rectilinear coordinate system to which we refer. That is,
, and are the magnitude projections of the
vector on the x, y and z axes, respectively.
Another way of representing a vector is by indicial notation.
If we let the subscript i denote any one of three
symbols x, y, z then 
represents symbolically a vector
since is merely an abbreviation for the set of symbols
, these being the scalar components of the
vector.
The indicial notation is most useful in dealing with tensors.
In this case we employ a second subscript j which can also
denote any one of the three symbols x, y, z, independent
of i. Thus, 
would represent a tensor in the
sense that it implies nine different permutations of the
subscripts x, y, z, taken two at a time. The possible
permutations lead to the nine terms:
which it is convenient to arrange in an orderly matrix array as indicated.
In using indicial notation, the convenction is adopted that when more than one of the same subscript is present in a given quantity, this indicates that the quantity is to be summed over all values of that subscript. For example, the product
represents a vector having the components
This convention is further illustrated in later sections.
It would be in order to review at this point some of the important vector operations which will be used in later developments, and to illustrated the equivalent operation in indicial notation.
The vector sum of and 
can be
denoted by
or, simply
The summation is represented graphically in Fig. 3.
The vector difference 
is the same as the vector sum of and
and can be represented by
or, simply
The scalar product of the vectors and
is denoted by 
.
The quantity is by definition a scalar having the
magnitude
Note that 
.
In indicial notation the scalar product would be
denoted simply as 
, where the summation over
all three values of i is implied. The scalar product can also be
represented as follows
where A and B are the
magnitudes of the vectors and ,
respectively, and 
is the angle between the two
vectors. Any two vectors define a plane containing
the two vectors and the angle is contained in this
plane.
The scalar product can be thought of as the magnitude of one times the magnitude of the normal projection of the other in the direction of the former (see Fig. 5).
In terms of the scalar product the magnitude of a given vector can be represented as
The vector product of two vectors and
is denoted by 
and is defined
as
If we expand the determinant by minors with respect to the top row as coefficients we have
The vector product is a vector. The magnitude can be expressed by the simple formula
where the bars denote absolute value or magnitude and
are the same as before.
Graphically, the magnitude expressed by (9)
represents the area of the parallelogram formed by the
vectors and in the plane of those
vectors (see Fig. 1.02-4 below).
The direction of the vector product is given by the right hand
rule convention. The direction of the vector
is perpendicular to the plane
of the vectors and and its sense is such
that it points in the direction of advance of a right hand screw if the
latter is turned
such that is turned toward through the angle
. See Fig. 1.02-5.
Note that 
The following properties of the scalar and vector products are very useful:
and conversely; and as long as neither nor
are null vectors,
and conversely.
The scalar triple product of vectors
can be expressed by
The value of this product is equal to the volume of the
parallelopiped formed by the three vectors 
.
This can be verified by reference to Fig. 8.
A vector triple product 
can be expressed in terms of Eq. 1.02 (7) as follows
where 
, 
, and 
are the components of
the vector product 
, i.e.
The scalar and vector products can also be represented symbolically
using indicial notation. At this point it would be well to repeat
the rule of addition: When more than one of the same
subscript appears in a given product or operation, this implies
that the quantity in question is to be summd over all values
of that subscript.
Thus the scalar product of and 
can be represented
simply by , which is merely an abbreviation
of
or
When dealing with a vector product, it is convenient to introduce
another subscript k which, like i and j, can take on the
values x, y, and z. Now if we let
represent the vector product of the vectors
and then we can write
where the convention adopted here is that the subscripts
i, j, k take on the values of x, y, z in cyclic order, i.e.
i, j, k can have the values x, y, z or y, z, x or
z, x, y, respectively.
Thus, the components of the vector product are