Electromagnetic Theory, Vol. 2 (1899) — part 11 of 31

SLBMBNT8 OF VSOTORIAL ALQEBBA AND ANALYSIS. 143

nuitics like winking, and carry out all instructionB made by the 4uithor.

If a vector a be multiplied by a scalar je, the result, written 2S or aoE^ is a vector x times as big as a, and having the same direction. Thus, if ftj be a unit vector parallel to a (or, more strictly, parallel to and concurrent with a), of unit length, we have a « oa^. 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. Fmr instance, the Cartesian coKnrdinatesa?, y, « of a point maybe retained. Thus, let r be the vector from the origin to any point, and let j, k be unit vectors ftom the origin along the three rectangular axes. We shall then have

x-xi, y-yj, a=2k, . . (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 4)rick. Also, the direction-cosines of r are x/r, y/r, s/r, and

. r^^a^-i-f + z'^, (2)

by Euclid I., 47.

The Addition of Vectors. Oircnital Property.

§ 105. This brings us to vector addition. The vector x signifies translation through the distance x in the direction L 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 a, 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 -comer of a brick, we may reach the opposite comer by three mutually perpendicular journeys along three edges of the brick. The final result is the same as if we went straight across from •comer to comer, that b, by carrying out the operation indicated by the vector r. This equivalence is expressed by

r-x+y-Hi, (3)

Or, by(l)b r=ici+yj+2k. . . • . . (4)

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The meaning of addition of veotors in this example is simply the carrying out of the operations implied by the individual Tectors added, the geometrical vector meaning a displaoement in space, or translation from one point to another. The order of addition is indifferent, since there are six ways of going from one comer 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.

Fio. 1. F|o. 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 figiue^ 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 vtd Q ; in the second case, with b first and then a» we go from P to R 1^ S. The final result in either case is equivalent to direct translation from P to R, symbolised by the vector &

Thus, c^a+bsb+a. (5)

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

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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.

Fio. 3. Fio. 4k

a+b+e+d in the figure. The final result of the Buocesslye tnuulations is always the same, viz., the direct translation A.

Or A»a+b+c+d. (6)

Thus any Tector A may be split up into the sum of any • number n of vectors, of which n—l are perfectly arbitrary, for instance a, b^ e in the figure. The remaining one d is, of oourse, not arbitrary. It is the reotor required to complete the circuit of yectors.

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 infoiitedmal 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 sero. That is, 2a »0, if a is the type'of the vectors summed* For a curved cucuit we shall have

/*-0, (7)

where dB is the vector element of the circuit. Here 8 itself may be taken to be the vector from any fixed point P to a point Q on the circuit, Fig. 4. Then cb is the infinitesimal

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

In a yector 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 ^ ihejittite change in the vectors 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 maybe regarded as reversing the tensor, without altering the direction: Uius

-a= - aaj = a X (- ai)==( - a) X a^ . . (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 eqiuvalent 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 dfouit in whatever order the addition be made, the matter may bedinched by means of 1, 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

a=i«i+jaj + ka3, J

b-i6i + j62+k63, V (11)

Now add up. All vectors parallel to 1 add in scalar fashion, for there is no change of direction. Similarly for j and k ; so we have

(a+b + c)-l(ai + 6i + Ci)+j(a, + 6, + c,)+k(a, + 65+cg.

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ELBMEKTS OF VSOTOBUL ALGEBRA AKD 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 -aide of an equation to the other are thus done identically as in the algebra of scalars. Addition has, it is true, not the same •meaning, but the vector meaning is not inconsistent with the •scalar meaning, and, in lact, includes the latter as a particular •case. There is never any conflict between + put between vectors and + put between scalars.

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

§ 106. In the above we have referred entirely to the geo- cnetrical vector. But a very important step further can be made referring to physical vectors, a step which inmiediately •does away with piles of ingeniously constructed and brain- wasting " demcmstrations," especially compiled for the tortur- ing of students and discouragement of learning. If a quantity ^ 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 jnultiplicatiou and other properties to be later considered. We may symbolise a physical vector by a straight line of given dength 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 epent in trying to make out Duchayla's proof, which was ■certainly elaborate and painstaking, though ben\imbing. 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 faiow that it follows the laws of the geometrical vector, the addition property of which does not want demonstrating, but •only needs pointing out, aa in the polygon of vectors ?

l2

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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 vectora a good deal of geometry can be simply done — ^better than by Euclid, a considerable part of whose 13* books consists of examples of how not to do it (especially Book v.). There is a Society for the Improyement 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 Veeton. Hotatioii and

Blustxationa.

§ 107. Coming next to the prodncts of vectors, it is to be- noted at the beginning that the ordinary idea ci 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 Aaaociatioa for the Improvemisiit of Qeometrical Teeohing. "It wm founded, we believe,, about ten years ago by a few teaohen^ who raalind that Euclid for

children ie, aa Mr. Heaviaide puts it, simply barbarous. Three pamphlets have been published by Messrs. Macmillan and Co. — a syllabus of plane geometry, corre:<ponding to Euclid, Books I. to VI. ; another of modem plane geuuietrj' ; and another of linear dynamics. Messrs. Swan, Souuen< scheiu and Co. have published an elementary geometrical conies, and there the labours of tke Association appear to have stopped. The Aasociaticm has had to struggle against the stubborn conventionalisin of the modem 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. iSmall as the vif^ible results of the Association have been, there is a distinct change of feeling taking place with regard to geometry, both as an ednoa^ tional subject and as an implement of soientifio w<n:k. At preeent geometey is taught as badly as Greek, even in the best public schools ; and the- educational value of Oreek is in many respects hij^er than that of EucUd."'

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ELEMENTS OF 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- Tuine this question by any prior reasoning of a legitimate nature. AVe 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 *no8t 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 410W immediately conoemed.

We d^isM the scalar product of a pair of veci^ors A and whose tensors are A and B, and whose included angle is ^, to <ie the scalar AB cos ^, and we denote it by AB. Thus

AB»ABcos^ (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 coper- 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

k«-l (13)

<with the convention borrowed from scalar algebra that ii is •equivalently denoted by i*. 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 As AAj, the square of any vector Is the same as the square of tta tensor ; or

A2 = A2 (U)

Also, the scalar product of any two parallel vectors is the product of their tensors, for ABs AB A^Bj, and A^B^ is now «nity.

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

y«0, jk = 0, kl = 0,

(15)

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ELEOTROMAONETIO XnEORT.

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That is, the scalar product of two perpendicular unit Tectors i» £ero. The same is, of course, true of any two perpendicular vectors. Thus the equation ABaQ means that A is perpen- dicular to B ; unless, indeed, one or other of them is zero. We have also

AiBi«cos(?, • • . . . (16)

hj dividing (12) by AB; or, the scalar product of any two unit vectors is the cosine of the included angle. This is reckoned positively from concurrent coincidence, so that as B goes from- 0 to 2r, A^B^ 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. Thus, 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 eitlier into the projection (perpendicularly) upon it of the other. It is the effective product, so to speak. In physical mathematics scalar products frecjuently 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 eti'eetive force. When the force and the velocity are perpendicular, the activity is nil, althoupjh the velocity may be changing — a fact which familiarity does not render less- strikincr. When F and v are parallel (whether concurrent or not), Fv becomes the same as Ff, 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-

ELBinCNTS OF VECTORIAL ALGEBRA AKD 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 refera, if B 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, the result, xa,h or ab^', is scalar, being simply x times ab. When multi- plied by a vector the result is a vector ; thus, cab 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. be is be times the vector a. But, instead of the dot, we may use brackets to indicate the same thing, thus cab 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 formuUw Thus, the electric stress vector on the plane whose unit normal ia N, is expressed by

KDN - N.^ED,

(equation (31), § 72) ; that is, the sum of two vectors, parallel to E the electric force and to N respectively, whose tensors are EBK and-}ED 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- pendicidar 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 he 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

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of scalar products, whioh will go far to justify them as working

utilities.

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

Now, in yeotor 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

A-a+b,

and B = c + d,

is AB»(a+b) (c+d) ) .

-ac+ad + bc+bdtr • • ' ^^^^

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 L According to our definition of a scalar product, the projection is AL 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

■T.KIIBNT8 OF VBCTOBIAL AJJOEBRk AKD ASALYSO, 153

«till with the same raalt algebraically. This is expressed by

Ai«(a+b + c)i':=al + bi + ci, . • . (19)

got by multiplyiiig (18) by i

If we now mtiltiply (19) by any scalar so that Bi->B, which is any Teotor, since i may have any direction, we obtain

AB»(a-fb+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 TeotOTB, say,

B-d+e+f (21)

If we substitute this in (20) we have

AB=a(d + e + f) + b(d + e + f) + c(d + e + f).

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

AB = (a + b + c) (d + e + f)

• 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-(Aii+Aaj+A8k)(Bil+BJ + B3k). . (22)

Now eft'ect the multiplications, remembering (13) and (15). The result is

AB = AiBi + 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

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the direotion cosines of the Tecton, 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- niay express the tensor at once in terms of the tensors and the cosines of the angles between a series of vectors into which the origiual vector is resolved. Thus, for two,

(A + B)a-.A2 + 2AB + B«, . . . (24) (A-B)««A«-2AB + B2; . . . (25> aud similarly for three,

(A + B + O)2 = A2 + B2 + C2 + 2AB + 2B0 + 2CA. (26)

Here (24) and (2.'i 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

(A + B)2 + (A-B)^ = 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)>-(A-B)s«4AB, . . . (28)

expressing the dilTerencc of the squares of the diagouals 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 vector* 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 diametei^ depending upon the oircumstaneeB of impact.

Beciprocal of a Vector.

§ 109. It id 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~^ or 1/a. Thus as a » aa^ we have

a-i-l = — = *i (29)

a aa^ a

Any unit vector is, therefore, its own reciprocal.

The reciprocal of a vector, being a vector, xnakes scalar pro- ducts with other vectors. Thus aV^ or a/b means the scalar product of a and \r\ and we therefore have

The tensor of a/b is (a/6) cos 0, 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~^ or a~^a equals unity.

In usinir reciprocals the defined meaning should be at- tended tu, especially when put in the fractional form. Thus we easily see that ab*^ = ab because the tensor of b is h- times the tensor of b^. But we cannot eijuivalently write a-/ab, because this is (ah) J cos 0, which is quite a different thing.

Notice that a"^ b-^ is not the same as (ab)"^. The first is- a-i 6"^ cos $, whilst the latter is a-^ b-^jcoa 6.

Expression of any Vector as the Sum of Three Independent

Vectors.

§ 110. We know that in the equation

r-arl + yj + j*,

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

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ELECTrwOMAGXETIC THEORY.

cn. ni,

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 <lo by multiplying by i. For i is perpendicular to j and k, so that multiplication by i gives

Similarly, rj^y and xk = z»

From this obvioiui case we can oonolude what to do when the veference Tecton are not perpendicular ; for instance, in

r«/a+^b + Ac, (31)

where a, b, c are any independent vectors. To find /we must multiply by a vector perpendicular to b and c, say If so that lb = 0, andlc^O. Then

rl»yid, therefore /«rl/aL

Similarly to isolate g and so that we get

r«^a + ^b + ??c (32)

al bm cn

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 case, 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 yectors bring us to the study of the vector product.

The Vector Product of Two Vectors. Blmrtrations.

§ 111. The auxiliary yectors 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

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ELEMENTS OF VECTORIAL ALGEBRA AND AKALTSIB. 157"

▼eetor, vis., the nonnal to the plane containing the given veotofs. Nor is the tensor of any oonscquence 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 auziliaiy vectors.

But the vector product of two vectors has a striotly-fizedf 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 aJ) sin d. Thus we define the vector product of two vectors a and b whose tensors are a and and whose included angle is to be a.

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

c-Vab, (33)

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

Similarly, V^ab may be used to denote the unit vector- parallel to Yab.

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

Fio. b.

Fio.6.

CBVgabso&sin 0,

(34)

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elbctbomaokbhc theory,

cn. nr.

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 aa to yary the angle -B, Starting from oolnddenoe, with ^«0, the tensor of e is sera It reaches a maximum (downwards) when ^ is a right angle, and falls to zero again when $ 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, e must be upwards from the paper.

The unit reference vectors 1, J, k are so arranged that when, in (33), a is i and b is J, then e 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

Vy-k, Vjk-1, Vki-j, . . (35)

t>ecau8e 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 a6 sin 0 and the scalar product ab is a/> cos ^, we have

(ah)2 + (Yah)2 = (ai)«. .... (38)

Oombinations of Three Vectora. The Farallelepipedal

Property.

§ 112. A vector product, being a vector, of course combines with other vectoi-s 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. .

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SLSHENT8 OF VBGTORIAL ALGEBRA AND ANALYSIS. 159

illostratiTO of notation, it may be mentioned that cVoab, ■^here, as before explained, Vgab Is the tensor, is obviously a vector parallel to o, but Y^h times as long. On the other hand, eY^ab^ where Yjab is the unit vector, is the same as •eVab/^oab; since by dividing by its tensor we unitise a vector. Similarly as regards VeViab. We may also have VjCVab and VjcVjab and VocVab, 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 dVaVbe, the scalar .product of d and the previously explained VaVbe.

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

eVab»yoabxccos<A, .... (39)

where (f> 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 Vo^-b 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 V^^ab, 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 .vidume, giving the equalities

aVbc = bVca = cVab, .... (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 Fimdamental Distributive Principle.

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

C = i(aj63-a86,)+j(ag6ji-ai6j) + k(ai6,-aj6i), . (41) in terms of i, J, k and the scalar components of a and b.

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BUOtBOXAONBnO THBORT.

OH. HI*.

To proye this, multiply (41) by a and b in turns to form the- Boalar products ae and be. Do this by the rule embodied in. equation (23). We get

ac « ai (0,63 - aj6,) + a, (ajb^ - Ojb^) + a, {ajf^ - 0,61),

be - 6i (a2&3 - ajJj) + h i^A - ^A) + h (^A - «2^i)-

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

ac»0, bc»0.

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

•   -  <hhy  +  {<hh  -  <hhy  +  (<hPi  -  (^lY-  (42) 
  

But this may, by common algebra, be transformed to

• (a^s + aj» + 038) (6i2 + + ^ ) - (a A (^3)

That is, c2«a2b2-(ab)a, (44)

or, c««a«6«-a«i«co8« ^-(a5sin 6)K

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 Yab or its negative.