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
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
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.
The following properties of the scalar and vector products are very useful:
and conversely; and as long as neither nor are null vectors,
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
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