CHAPTER III.

VECTOR MULTIPLICATION AND DIVISION.

15. We trust we have made the reader understand by what we stated in our Introductory Chapter, that, whilst we retain for 'multiplication' all its old properties, so far as it relates to ordinary algebraical quantities, we are at liberty to attach to it any signification we please when we speak of the multiplication of a vector by or into another vector. Of course the interpretation of our results will depend on the definition, and may in some points differ from the interpretation of the results of multiplication of numerical quantities.

It is necessary to start with one limitation. Whereas in Algebra we are accustomed to use at random the phrases 'multiply by' and 'multiply into' as tantamount to the same thing, it is now impossible to do so. We must select one to the exclusion of the other. The phrase selected is 'multiply into'; thus we shall understand that the first written symbol in a sequence is the operator on that which follows: in other words that aß shall read 'a into ẞ', and denote a operating on ß.

16. As in the Cartesian Geometry, so here we indicate the position of a point in space by its relation to three axes, mutually at right angles, which we designate the axes of x, y, and z respectively. For graphic representation the axes of x and y are drawn in the plane of the paper whilst that of z being perpendicular to that plane is drawn in perspective only. As in ordinary

X

geometry we assume that when vectors measured forwards are represented by positive symbols, vectors measured backwards will be represented by the corresponding negative symbols. In the figure before us, the positive directions are forwards, upwards and outwards; the corresponding negative directions, backwards, downwards and inwards.

With respect to vector rotation we assume that, looked at in perspective in the figure before us, it is negative when in the direction of the motion of the hands of a watch, positive when in the contrary direction. In other words, we assume, as is done in modern works on Dynamics, that rotation is positive when it takes place from y totz, z to x, x to y: negative when it takes place in the contrary directions (see Tait, Art. 65).

Unit vectors at right angles to each other.

17. DEFINITION. If i, j, k be unit vectors along Ox, Oy, Oz respectively, the result of the multiplication of i into j or ij is defined to be the turning of j through a right angle in the plane perpendicular to i and in the positive direction; in other words, the operation of i on j turns it round so as to make it coincide with k; and therefore briefly ij=k.

To be consistent it is requisite to admit that if i instead of operating on j had operated on any other unit vector perpendicular to i in the plane of yz, it would have turned it through a right angle in the same direction, so that ik can be nothing else than -j. Extending to other unit vectors the definition which we have illustrated by referring to i, it is evident that j operating on k must bring it round to i, or jk = i.

Again, always remembering that the positive directions of rotation are y to z, z to x, x to y, we must have ki=j.

18. As we have stated, we retain in connection with this definition the old laws of numerical multiplication, whenever numerical quantities are mixed up with vector operations; thus 2i. 3j6ij. Further, there can be no reason whatever, but the contrary, why the laws of addition and subtraction should undergo

T. Q.

3

any modification when the operations are subject to this new definition; we must clearly have

Finally, as we are to regard the operations of this new definition as operations of multiplication-magnitude and motion of rotation being united in one vector symbol as multiplier, just as magnitude and motion of translation were united in one vector symbol in the last chapter-we are bound to retain all the laws of algebraic multiplication so far as they do not give results inconsistent with each other. In no other way can the conclusions be made to compare with those deduced from the corresponding operations in the previous science. Thus we retain what Sir William Hamilton terms the associative law of multiplication: the law which assumes that it is indifferent in what way operations are grouped, provided the order be not changed; the law which makes it indifferent whether we consider abc to be a x bc or ab × c. This law is assumed to be applicable to multiplication in its new aspect (for example that ÿjk = ij. k), and being assumed it limits the science to certain boundaries, and, along with other assumed laws, furnishes the key to the interpretation of results.

The law is by no means a necessary law. Some new forms of the science may possibly modify it hereafter. In the meantime the assumption of the law fixes the limits of the science.

The commutative law of multiplication under which order may be deranged, which is assumed as the groundwork of common algebra (we say assumed advisedly) is now no longer tenable. And this being the case it is found that the science of Quaternions breaks down one of the barriers imposed by this law and expands itself into a new field.

ij is not equal to ji, it is clearly impossible it should be.

A simple inspection of the figure, and a moment's consideration of the definition, will make this plain. The definition imposes on i as an operator on j the duty of turning j through a right angle as if by a left-handed turn with a cork-screw handle, thus throwing jup from the plane xy; when, on the other hand, j is the operator

and i the vector operated on, a similar left-handed turn will bring i down from the plane of xy. In fact ijk, ji-k, and so ij=-ji.

19. We go on to obtain one or two results of the application of the associative law.

or

1. Since ijk, we have i . ij = ik=—j.

Now by the law in question,

i . ÿj = ïï . j = ï3 . j ;

.. i.j=-j,

i=-1.

Our first result is that the square of the unit vector along Ox is - 1; and as Ox may have any direction whatever, we have, generally, the square of a unit vector -1. In other words, the repetition of the operation of turning through a right angle reverses a vector.

or no change is produced in the product so long as direct cyclical order is maintained.

3. But ikji. kji.i=—¿2=+1;

.. ÿjk=—ikj,

or a derangement of cyclical order changes the sign of the product. This last conclusion is also manifest from Art. 18.

Vectors generally not at right angles to each other.

20. We have already (Art. 8) laid down the principle of separation of the vector into the product of tensor and unit vector; and we apply this to multiplication by the considerations given in Art. 18, from which it follows at once that if a be a vector along Ox containing a units, ẞ a vector along Oy containing 6 units,

b

a = ai, ẞ=bj, and aß= abij.

or the square of a vector is the square of the corresponding line with the negative sign.

Seeing therefore the facility with which we can introduce tensors whenever wanted, we may direct our principal attention, as far as multiplication is concerned, to unit vectors.

21. We proceed then next to find the product aß, when a and ẞ are vectors not at right angles to one another.

vector OA = OM + MA= OM + ON (Art. 1)
= part of OB + part of OC (Art. 3).

M B

Now it is evident that OM as a line is that part of OB which is represented by the multiplier cos 0, or

similarly that ON- OC sin 0: applies to them as vectors; i. e.

and

But

vector OM=ẞ cos 0,

consequently

OM = OB cos 0, and

(Art. 3) the same

vector ON = y sin 0;

.. a=ẞ cos 0+ y sin 0,

aß = (ẞ cos 0 + y sin 0) ß

[Observe that y, ẞ and

to j, i and -k of Art. 17.]

of the present Article correspond

.. aß= cos 0 + € sin 0.