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

Mon Dec 1 08:50:29 CST 1997