Electromagnetic Theory, Vol. 1 (1893) — part 11 of 31

matics like winking, and carry out all instructions made by the author.

If a vector a be multiplied by a scalar x, the result, written afa or az, is a vector x times as big as a, and having the same direction. Thus, if a3 be a unit vector parallel to a (or, more strictly, parallel to and concurrent with a), of unit length, we have a = aar It is sometimes useful to separately represent the direction and length of a vector, and the above is a con- venient way of doing it without introducing new letters. This applies to any vector. But it need not be an absolute rule. For instance, the Cartesian co-ordinates xt y, z of a point may be retained. Thus, let r be the vector from the origin to any point, and let i, j, k be^ unit vectors from the origin along the three rectangular axes. We shall then have

x = ^i, y = yj, z = zk, . . (1)

where x is the vector projection of r on the i axis, as we know by the elementary geometry of a rectangular parallelepiped or brick. Also, the direction-cosines of r are x/r, yjr, z/r, and

by Euclid I., 47.

The Addition of Vectors. Circuital Property.

§ 105. This brings us to vector addition. The vector x signifies translation through the distance x in the direction i. If, now, after performing this operation, we carry out the operation indicated by y, viz., translation through the distance y in the direction j ; and, lastly, carry out the operation z, or translation through the distance z in the k direction, we shall arrive at the end of the vector r. That is, starting from one corner of a brick, we may reach the opposite corner by three mutually perpendicular journeys along three edges of the brick. The final result is the same as if we went straight across from corner to corner, that' is, by carrying out the operation indicated by the vector r. This equivalence is expressed by

r = x + y + z, . . , . , . (3) Or, by (1), r = XL + yj + zk. 3 0 , , . (4)

144

ELECTROMAGNETIC THEORY.

CH. III.

The meaning of addition of vectors in this example is simply the carrying out of the operations implied by the individual vectors added, the geometrical vector meaning a displacement in space, or translation from one point to another. The order of addition is indifferent, since there are six ways of going from one corner to the opposite one of a brick along its edges.

We see that any vector may be expressed as the sum of three mutually perpendicular vectors, viz., its vector projections on the axes. Furthermore, by the use of a skew parallelepiped Instead of a brick we see that any vector may be expressed as the sum of three other vectors having any directions we please, provided they are independent, or not all in the same plane. For in the latter case the parallelepiped degenerates to a plane figure.

FIG. 2.

But it is perhaps best to explain vector addition in general without any reference to axes. Thus let a and b be vectors to be added, a meaning translation from P to Q in the first figure, or through an equal distance along any parallel line, as from S to R in the second figure, whilst b means translation from Q to R in the first figure, or through an equal distance along any parallel line, as from P to S in the second figure. In the first case, performing the operation a first, and then b, we go from P to R irid Q ; in the second case, with b first and then a, we go from P to R vid S. The final result in either case is equivalent to direct translation from P to R, symbolised by the vector c.

Thus,

(5)

The above may be extended to any number of vectors. Or we may reason thus : — Let A be any given vector, translating, say,

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 145

from P to Q. We need not go from P to Q direct, but may follow any one of an infinite number of paths, as for example,

I

f

P A

FIG. 3.

Q

FIG. 4.

a + b + c-fdin the figure. The final result of the successive translations is always the same, viz., the direct translation A.

Or A=a+b+c+d (6)

Thus any vector A may be split up into the sum of any number n of vectors, of which n — 1 are perfectly arbitrary, for instance ,a, b, c in the figure, The remaining one d is, of course, not arbitrary. It is the vector required to complete the circuit of vectors.

The vectors need not be in one plane. Nor need the path followed consist of finite straight portions. It may be wholly or partly curved. The curved portions are then made up of infinitesimal vectors. Each curved portion may be replaced by the vector joining its terminals.

Since the original vector and the substituted vectors form a circuit, if the positive direction in the circuit be the same for all the vectors, we may express vector addition thus: — The sum of any number of vectors which make a circuit is zero. That is, 2a = 0, if a is the type of the vectors summed. For a curved circuit we shall have

(7)

where da is the vector element of the circuit. Here s itself may be taken to be the vector from any fixed point P to a point Q on the circuit, Fig. 4. Then ds is the infinitesimal

146 ELECTROMAGNETIC THEORY. CH. III.

change in s made in an infinitesimal step along the circuit, that is, it is the vector element of the circuit itself.

In a vector equation every term is a vector, of course, how- ever the individual terms may be made up, and every vector equation expresses the fact symbolised by (7), or by 2a = 0 when the vectors are finite. In the latter case, a may be also regarded as As, the finite change in the vector s from any fixed origin produced by passing from beginning to end of the vector a.

The - sign prefixed to a vector is the same as multiplication by - 1, and its effect is simply to reverse the direction of trans- lation. Or it may be regarded as reversing the tensor, without altering the direction: thus

• a = - aax = a x ( - ax) = ( - a) x aj . . (8). Thus, in any vector equation, for example

a + b + c + d + e + f=0, . . . . (9)

we may transfer any terms to the other side by prefixing the

• sign. Thus,

a + b + c= -d-e-f- -(d + e + f). . . (10)

In the form (9) we express the fact that translation in a circuit is equivalent to no translation. In (10), however, we express the equivalence of two translations by different paths from one point to another.

If any trouble be experienced in seeing the necessary truth of (9) for a circuit in whatever order the addition be made, the matter may be clinched by means of i, j, k, the unit rectangular vectors ; as we saw before, the resolution of vectors into rectangular component vectors depends only upon the properties of the right angled triangle. Thus split up the vectors a, b, c into components, we have

k63, ..>.';. (11) c = icj + jc2 + kc3, ) .

Now add up. All vectors parallel to i add in scalar fashion, for there is no change of direction. Similarly for j and k ; so we have

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 147

Now the scalar additions may be done in any order we please. It follows that the same is true in vector addition. The addition and subtraction of vectors and transfer from one side of an equation to the other are thus done identically as in the algebra of scalars. Addition has, it is true, not the eame meaning, but the vector meaning is not inconsistent with the scalar meaning, and, in fact, includes the latter as a particular case. There is never any conflict between + put between vectors and + put between scalars.

Application to Physical Vectors. Futility of Popular Demon- strations. Barbarity of Euclid.

§ 106. In the above we have referred entirely to the geo- metrical vector. But a very important step further can be made referring to physical vectors, a step which immediately does away with piles of ingeniously constructed and brain- wasting " demonstrations," especially compiled for the tortur- ing of students and discouragement of learning. If a quantity be recognised to be a directed magnitude, it is, in its mathe- matical aspect, a vector, and is therefore subject to the same laws as the vector, or geometrical vector. So, just as we have the triangle, or parallelogram, or polygon of geometrical vectors, we must have the same property exemplified in the addition of all vectors, as velocity, acceleration, force, &c. This identity of treatment applies not only to the addition property, but to the multiplication and other properties to be later considered. We may symbolise a physical vector by a straight line of given length and direction.

It used to be thought necessary to give demonstrations of the parallelogram of forces, perhaps even before the student knew what force meant. I have some dim recollection of days spent in trying to make out Duchayla's proof, which was certainly elaborate and painstaking, though benumbing. Max- well, in his treatise, elaborated a demonstration that electric currents compounded according to the vector law. But surely there is something of the vicious circle in such demonstrations. Is it not sufficient to recognise that a quantity is a vector, to know that it follows the laws of the geometrical vector, the addition property of which does not want demonstrating, but only needs pointing out, as in the polygon of vectors ?

L2

148 ELECTROMAGNETIC THEORY. CH. IIL

Although this is not the place for exercises and examples, yet it is worth while to point out that by means of the addition property of vectors a good deal of geometry can be simply done — better than by Euclid, a considerable part of whose 12 books consists of examples of how not to do it (especially Book V.). There is a Society for the Improvement of Geometrical Teaching. I have no knowledge of its work ; but as to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then, not Euclid, but algebra. Next, not Euclid, but practical geometry, solid as well as plane ; not demonstrations, but to make acquaintance. Then, not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. This covers a large ground. When more advanced, bring in the vector product. Elementary calculus should go on simultane- ously, and come into the vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous.*

The Scalar Product of Two Vectors. Notation and

Illustrations.

§ 107. Coming next to the products of vectors, it is to be noted at the beginning that the ordinary idea of a product in

• From The Electrician, December 4, 1891, p. 106, I learn that the correct title of the society above alluded to is the Association for the Improvement of Geometrical Teaching. " It was founded, we believe, about ten years ago by a few teachers, who realised that Euclid for children is, as Mr. Heaviside puts it, simply barbarous. Three pamphlets have been published by Messrs. Macmillan and Co. — a syllabus of plane geometry, corresponding to Euclid, Books I. to VI. ; another of modern plane geometry ; and another of linear dynamics. Messrs. Swan, Sonnen- schein and Co. have published an elementary geometrical conies, and there the labours of the Association appear to have stopped. The Association has had to struggle against the stubborn conventionalism of the modern schoolmaster, who pleads that he cannot make any change, because of the Universities. After a considerable fight, Mr. Hamblin Smith's common- sense proofs of Euclid's problems were accepted by Cambridge examiners. Small as the visible results of the Association have been, there is a distinct change of feeling taking place with regard to geometry, both as an educa- tional subject and as an implement of scientific work. At present geometry is taught as badly as Greek, even in the best public schools ; and the educational value of Greek is in many respects higher than that of Euclid."

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 149

arithmetic and in scalar algebra does not apply to vectors, because they are not scalars. We cannot, therefore, say before- hand what the product of two vectors ought to be, or deter- mine this question by any prior reasoning of a legitimate nature. We must examine in what way vectors enter into combination as products, and then introduce definitions and conventions of notation to give expression to the facts in the simplest and most convenient way. After this, the work is deductive.

By this examination we are led to recognise two distinct kinds of products of a pair of vectors, the scalar product and the vector product. It is with the former of these that we are now immediately concerned.

We define the scalar product of a pair of vectors A and B whose tensors are A and B, and whose included angle is 6, to be the scalar AB cos 0, and we denote it by AB. Thus

AB = ABcos0 (12)

defines the scalar product and its notation.

To see its full significance and how it works out, let us first apply (12) to the unit reference vectors, i, j, k, which are oeper- pendicular, by taking A and B to be one or other of them in turn. First multiplying each of i, j, k by itself, we get

12 = 1, J2 = l, k2=l, . . . . (13)

with the convention borrowed from scalar algebra that ii is equivalently denoted by i2. This may be called the square of i. In each of the three cases (13) the vectors multiplied together are of unit length, and are parallel and concurrent, so that the scalar product, according to (12), is unity.

Similarly, the square of any unit vector is unity. And since A = AA15 the square of any vector is the same as the square of its tensor ; or

A2 = A2 (14)

Also, the scalar product of any two parallel vectors is the product of their tensors, for AB = AB A1B1, and A1B1 is now unity.

Next make scalar products of i, j, k in pairs. We get

ij = 0, jk = 0, ti = 0 (15)

150 ELECTROMAGNETIC THEORY. CH. III.

That is, the scalar product of two perpendicular unit vectors is zero. The same is, of course, true of any two perpendicular vectors. Thus the equation AB = 0 means that A is perpen- dicular to B ; unless, indeed, one or other of them is zero. We have also ^

AA-cosfl, (16)

by dividing (12) by AB ; or, the scalar product of any two vectors is the cosine of the included angle. This is reckom positively from concurrent coincidence, so that as 0 ge2s from 0 to 2 TT, AjBi goes from 1 through 0 to — 1, and then through 0 again to + 1.

There is, strictly, no occasion to introduce trigonometry. Or we might make the trigonometry be a simultaneously developed subject. It is, in fact, a branch of vectorial algebra, being scalar developments of parts thereof. We may employ the idea of perpendicular projection simply. rrThus, we may say that the scalar product of a pair of unit vectors is the length of the projection of either upon the other ; and that the scalar product of any vector A and a unit vector i is the projection of A upon the axis of i ; and, comprehensively, that the scalar product AB of any two vectors A and B is the product of the tensor of either into the projection (perpendicularly) upon it of the other. It is the effective product, so to speakH In physical mathematics scalar products frequently have reference to energy, or activity, or connected quantities. Thus, if F be a force and v the velocity of its point of application, their scalar product Fv is the activity, of the force ; it is the product of the speed and the effective force. When the force and the velocity are perpendicular, the activity is nil, although the velocity may be changing — a fact which familiarity does not render less striking. When F and v are parallel (whether concurrent or not), Fv becomes the same as Fi;, in the common meaning of a product. • A notation^that harmonises in this way is obviously a convenient one.

Notice, also, that AB = BA, as in scalar algebra.

A frequently occurring operation is the surface integral of the normal component of a vector ; for example, to express the amount of induction through a surface. Here the idea of a unit normal vector N is useful. The normal component of B

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 151

is then NB, and: the integral is 2 NB, the summation extending over the surface.

Similarly, to express the line-integral of the effective component of a vector along a line, we may let T be the unit element of curve ; that is, the unit tangent ; then TE is the tangential component of the vector E, and 2TE is the integral ; for example, the voltage between two points along the path to which the summation refers, if E be the electric force.

Since ab is a scalar, it behaves as a scalar, when considered as a whole. Thus, when multiplied by a scalar, x, the result, #ab or aba:, is scalar, being simply x times ab. When multi- plied by a vector the result is a vector ; thus, c.ab or ab.c means ab times the vector c . The dot here acts rather as a separator than as a sign of multiplication. Thus, to illustrate, ca.b means ca times the vector b ; and, similarly, a.bc is be times the vector a. But, instead of the dot, we may use brackets to indicate the same thing, thus c.ab may be written c(ab). This is, perhaps, preferable in initiatory work, but I think the dot plan is more generally useful.

We have an example of this combination of three vectors in the stress formulae. Thus, the electric stress vector on the plane whose unit normal is N, is expressed by

E.DN - N.JED,

(equation (31), § 72) ; that is, the sum of two vectors, parallel to E the electric force and to N respectively, whose tensors are EDN and -JED respectively, where D is the displacement. The interpretation as a tension along E combined with an equal lateral pressure, obtained by taking N parallel to and then per- pendicular to E or D, has been already discussed.

Fundamental Property of Scalar Products, and Examples.

§ 108. As all the preceding is involved in the definition of a scalar product, and is obvious enough, it may be regarded merely as illustrative. The reader might, in fact, say that ha knew it all before in one form or another, trigonometrical or geometrical, and that he did not see any particular advantage in the way of stating it in the notation of scalar products. But we now come to a very striking and remarkable property

152 ELECTROMAGNETIC THEORY. CH. 111.

of scalar products, which will go far to justify them as working utilities.

We know that in scalar algebra, when we have a product xyt we may express x by the sum of any number of other quantities, and similarly as regards y, and then obtain the complete product xy by adding together all the component products obtained by multiplying every element of x into every element of y ; and that this process may be carried out in any order.

Now, in vector algebra, we know already that there is a partial similarity, viz., that we can decompose a vector A into the sum of any number of others, and similarly as regards B. The question now is whether the scalar product AB is the sum of all the scalar products made up out of the components of A paired with the components of B, taken in any order. For example, if

and B = c + d,

is AB = a + bc + d )

J *

The answer is Yes, and the process of manipulation is the same as in scalar algebra, that is, as if all the vectors were scalars. Moreover, this property admits of demonstration in a sufficiently simple manner as to enable one to see its truth.

Start with the vector A, and first split it up into

A = a + b + c, ..... (18)

two or three vector components being sufficient for illustration. Now project the vector A perpendicularly upon any axis, say that of i. According to our definition of a scalar product, the projection is Ai. ' Now it requires no formal demonstration, but becomes evident as soon as the meaning of the proposition is correctly conceived, that the projection of A upon any axis is the same as the sum of the projections of its component vectors, a, b, c, on that axis. That is, the latter projections are either all positive or all negative, and fit together to make up the projection of A ; or else some may be negative and others positive, when there is overlapping and cancelling, but

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 153

«till with the same result algebraically. This is expressed by Ai = (a + b + c)i = ai + bi + ci, . . . (19)

;got by multiplying (18) by i.

If we now multiply (19) by any scalar B, so that Bi=B, which is any vector, since i may have any direction, we obtain

AB = (a + b + c)B = aB + bB + cB, . . (20)

the same as if we multiply (18) by B direct.

Similarly, we may split up B into the sum of any number of other vectors, say,

(21)

If we substitute this in (20) we have

Now make use of the same reasoning which established (19) and (20), applied to the bracketed vectors, and we establish •the property fully, with the result

= ad + ae + af + bd + be + bf + cd + ce + cf ;

the expansion being done formally as in scalar algebra in every respect ; since the various terms may be written in any order, and each may be reversed, thus, ad = da.

I have already remarked that a good deal of geometry may be done by the addition property alone. The range of applica- tion is greatly extended by the use of the scalar product and the fundamental property (17).

To obtain the Cartesian form of AB, put the vectors in terms of i, j, k; thus

AB = (A1i + A2j+A3k) (B^ + BJ + Bak). . (22)

Now effect the multiplications, remembering (13) and (15). The result is

AB = A1B1 + A2B2 + A3B3. . . . (23)

For example, the activity of a force is the sum of the activities of its component forces in any three coperpendicular directions. Or, if A and B are unit vectors, we express the cosine of the angle between them in terms of the products of

154 ELECTROMAGNETIC THEORY. CH. 111-

the direction cosines of the vectors, each product referring to one axis. In (23) we may, if we please, vectorise the six scalars on the right side.

Since the square of a vector is the square of its tensor, we may express the tensor at once in terms of the tensors and the cosines of the angles between a series of vectors into which -the original vector is resolved. Thus, for two,

(24)

(A-B)2 = A2-2AB + B2; . . . (25) and similarly for three,

(26)

Here (24) and (25) apply to a parallelogram, and (26) to a parallelepiped. In (24), (25), if A and B be the vector sides of a parallelogram, then (A + B) and (A-B) are the vector diagonals ; so (24) gives the length of one diagonal and (25} that of the other.

Adding (24) and (25) we obtain

-B)2 = 2(A2 + B2), . . (27)

expressing that the sum of the squares of the diagonals equals the sum of the squares of the four sides. Similarly, by sub- traction

(A + B)2-(A-B)2 = 4AB, . . . (28)

expressing the difference of the squares of the diagonals as four times the scalar product of two vector sides.

Equation (27) also shows that the sum of the squares of the distances of the ends of any diameter of a sphere from a fixed point is constant. For if A is the vector from the fixed point to the sphere's centre, and B the vector from the centre to one end of the diameter, then A + B and A-B are the vectors- from the fixed point to the ends of the diameter. Whence, by (27), the proposition.

There is an application of this in the kinetic theory of gases* For when two elastic spheres collide they keep the sum of their kinetic energies constant. If, then, their velocities before collision be A + B and A-B, their velocities after collision are indicated by vectors from the origin (in the velocity diagram)

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 155

to the ends of some other diameter of the sphere described upon the line joining their original positions (in the velocity diagram) as diameter ; which is actually the new diameter depending upon the circumstances of impact.

Reciprocal of a Vector.

§ 109. It is occasionally useful to employ the reciprocal of a vector in elementary vector algebra. We define the reciprocal of a vector a to be a vector having the same direction as a, and whose tensor is the reciprocal of that of a. We may denote the reciprocal of a by a-1 or I/a. Thus as a = aap we have

(29)

Any unit vector is, therefore, its own reciprocal.

The reciprocal of a vector, being a vector, makes scalar pro- ducts with other vectors. Thus ab-1 or a/b means the scalar product of a and b-1, and we therefore have

i-^-S-H-S- ' • • (30)

The tensor of a/b is (a/b) cos 6, where 6 is the angle between a and b, or between their reciprocals, 'or between either and the reciprocal of the other.

So a/a or aa'1 or a'1 a equals unity.

In using reciprocals the defined meaning should be at- tended to, especially when put in the fractional form. Thus we easily see that aV1 = ab/62, because the tensor of b is b2 times the tensor of b'1. But we cannot equivalently write a2/ab, because this is (a/6)/cos 0, which is quite a different thing.

Notice that a"1 b-1 . is not the same as (ab)-1. The first is a-1 6'1 cos 6, whilst the latter is or1 ft^/cos 0.

Expression of any Vector as the Sum of Three Independent

Vectors.

§ 110. We know that in the equation

i yd -i »"^"j

where r is the vector distance of a point from the origin, the scalars are the lengths of the projections of r upon the axes.

156 ELECTROMAGNETIC THEORY. CH. III.

How should we, however, find them in terms of r algebraically ? To find x we must operate on the equation in such a manner as to cause the j and k terms to disappear. Now this we can do by multiplying by i. For i is perpendicular to j and k, so that multiplication by i gives

Similarly, rj = y and rk = z.

From this obvious case we can conclude what to do when the reference vectors are not perpendicular ; for instance, in

...... (31)

where a, b, c are any independent vectors. To find / we must multiply by a vector perpendicular to b and c, say 1, so that Ib = 0, and Ic = 0. Then

rl=/al, therefore /=rl/al Similarly to isolate g and A, so that we get

'-i'OO ..... (32>

where 1, m, n are vectors normal to the three planes of b,c, c,a and a,b. This exhibits explicitly the expansion of any vector in terms of any three independent vectors a, b, c, as the three edges of a skew parallelepiped. This ease, of course, reduces to the preceding Cartesian case by taking a, b, c to be i, j, k, when 1, m, n will also be i, j, k, or any scalar multiples of the same.

Observe the peculiarity that auxiliary vectors are used, each of which is perpendicular to two others, that is, to the plane containing them. These auxiliary vectors bring us to the study of the vector product.

The Vector Product of Two Vectors. Illustrations.

§ 111. The auxiliary vectors just employed in the expansion of a vector into the sum of three vectors having any independent directions, are examples of vector products. Two vectors being given, their vector product is perpendicular to both of them. Of course, disregarding magnitude, there is but one such

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS.

157

vector, viz., the normal to the plane containing the given vectors. Nor is the tensor of any consequence in the example in question, for it will be observed that one of the vectors, 1, m, n, appears both in the numerator and in the denomi- nator of one of the three fractions in equation (32), so that the values of the fractions are independent of the tensors of the auxiliary vectors.

But the vector product of two vectors has a strictly-fixed tensor, depending upon those of the component vectors and their inclination. There is particular advantage in taking the tensor of the vector product of a and b to be ab sin 0. Thus we define the vector product of two vectors a and b whose tensors are a and &, and whose included angle is 0, to be a

b

FIG. 5.

FIG. 6.

third vector c whose tensor c equals ab sin 0, and whose direc- tion is perpendicular to the plane of a and b ; the positive direction of c being such that positive or right-handed rotation about c carries the vector a to b. This vector product is denoted by

c = Vab, ...... (33)

and its tensor may be denoted by V0ab, so that we have

(34)

Similarly, Vxab may be used to denote the unit vector parallel to Vab.

The only troublesome part is to correctly fix which way along the perpendicular to the plane of a and b is to be considered positive. Examples will serve to clinch the matter. Thus, let

158 ELECTROMAGNETIC THEORY. CH. III.

a be towards the north, and b towards the east on the earth's surface ; then c is straight downwards. Again, in Fig. 5, the direction of c is downwards through the paper, and its tensor is the area of the parallelogram upon a and b.

Let b be fixed, whilst a moves round so as to vary the angle Q. Starting from coincidence, with 6 = 0, the tensor of c is zero. It reaches a maximum (downwards) when 0 is a right angle, and falls to zero again when 0 reaches two quadrants, and a is in the same line with b, though discurrent. After this, in the next two quadrants, the same numerical changes are gone through ; but now, the sine being negative, c must be upwards from the paper.

The unit reference vectors i, j, k are so arranged that when, in (33), a is i and b is j, then c is k ; noting in Fig. 6, that k is supposed to go downwards, or away from the reader. Thus, in accordance with our definition we have

Vij = k, Vjk = i, Vki=j, . . (35)

because the mutual angles are quadrants and the tensors unity. Observe the preservation of the cyclical order i, j, k in (35). Also, we have

Vii = 0, Vjj = 0, Vkk = 0, . . (36)

because the vectors paired are coincident.

By the definition, the reversal of the order of the letters in a vector product negatives it. Thus .

Vab= -Vba. ..... (37)

Since the tensor of Vab is ab sin 0 and the scalar product .ab is ab cos 0, we have

(38)

Combinations of Three Vectors. The Parallelepipedal Property.

§ 112. A vector product, being a vector, of course combines with other vectors to make scalar and vector products again. Thus cVab, where c is any new vector, means the scalar product of c and Vab; and VcVab means the vector product of c and Vab. These are both important combinations, which occur frequently, and their interpretations and expansions will be given presently.

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 159

As illustrative of notation, it may be mentioned that cV0ab, where, as before explained, V0ab is the tensor, is obviously a vector parallel to c, but V0ab times as long. On the other •hand, cVjab, where Vxab is the unit vector, is the same as cVab/V0ab; since by dividing by its tensor we unitise a vector. Similarly as regards VcVjab. We may also have V^Vab and V^V^b and V0cVab, and various other modifica- tions, whose meanings follow from the definition of a vector product and its notation.

We do not often go further in practical vector algebra than combinations in threes ; for instance, on to dVaVbc, the scalar product of d and the previously explained VaVbc.

The scalar product cVab has an important geometrical illus- tration. Its value is given by

cVab = V0abxccos<£, .... (39)

where <£ is the angle between c and Vab. This is by the defi- nition of the scalar product, and of the tensor of a vector.

Now refer to Fig. 5 again. We know that V0ab is the area of the parallelogram. We also know that the volume of a parallelepiped is the product of its base and altitude. Construct, •then, a parallelepiped whose three edges meeting at a corner are a, b, and c. The area of one of its bases is V0ab, and the corresponding altitude is c cos <£. Therefore, by (39), cVab is the volume of the parallelepiped.

But there are two other bases and two other altitudes to correspond, so there are two other ways of expressing the 'volume, giving the equalities

.... (40)

in which observe the preservation of cyclical order, done to keep the sign the same throughout, as will be verified a little later.

Semi-Cartesian Expansion of a Vector Product, and Proof of the Fundamental Distributive Principle.

§ 113. The semi-Cartesian expansion of c = Vab is

C = i(a263-a362)+j(a361-a163)+k(a162-a26l), . (41) in terms of i, j, k and the scalar components of a and b.

160 ELECTROMAGNETIC THEORY. OH. Ill*

To prove this, multiply (41) by a and b in turns to form the scalar products ac and be. Do this by the rule embodied in equation (23). We get

ac = aj (a263 be = 6j (a26;i

But, by cancelling, all the terms on the right disappear, That is,

ac = 0, be = 0.

From these we know that c is perpendicular to a and to b. It is, therefore, Vab itself, or a multiple of the same. To find its tensor, square (41). We get

c2 = (aj), - 0362)8 + (ajh _ aij3)2 + (ai&2 _ a2ji)a. (42) But this may, by common algebra, be transformed to

c2 = fa* + a* + a32) (V + Z>22 + &32) - (a^ + a2&2 + «353)2. (43) That is, c2 = a2b2- (ab)2, ..... (44)

or, c2 = a262 - a262 cos2 6 = (ab sin 0)2.

The tensor c is, therefore, that of Vab itself, or else its negative. Equation (44) is the same as (38), in a slightly different form. Equation (41) is proved, except that it remains to be seen whether the right member represents Vab or its negative.