PlanetPhysics/Vectors and Their Composition

Chapter 1: Vectors and Their Composition
From An Elementary Treatise On Quaternions by Peter Guthrie Tait.

1. FOR at least two centuries the geometrical representation of the negative and imaginary algebraic quantities, $$-1$$ and $$\sqrt{-1}$$ has been a favourite subject of speculation with mathematicians. The essence of almost all of the proposed processes consists in employing such expressions to indicate the DIRECTION, not the length, of lines.

2. Thus it was long ago seen that if positive quantities were measured off in one direction along a fixed line, a useful and lawful convention enabled us to express negative quantities of the same kind by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics.

3. Wallis, towards the end of the seventeenth century, proposed to represent the impossible roots of a quadratic equation by going out of the line on which, if real, they would have been laid off. This construction is equivalent to the consideration of $$\sqrt{-1}$$ as a directed unit-line perpendicular to that on which real quantities are measured.

4. In the usual notation of Analytical Geometry of two dimensions, when rectangular axes are employed, this amounts to reckoning each unit of length along $$Oy$$ as $$+\sqrt{-1}$$, and on $$Oy'$$ as $$-\sqrt{-1}$$; while on $$Ox$$ each unit is $$+1$$, and on $$Ox'$$ it is $$-1$$. T. Q. I.

If we look at these four lines in circular order, i.e. in the order of positive rotation (that of the northern hemisphere of the earth about its axis, or opposite to that of the hands of a watch), they give

$$1, \sqrt{-1}, -1, -\sqrt{-1}.$$

In this series each expression is derived from that which precedes it by multiplication by the factor $$\sqrt{-1}$$. Hence we may consider $$\sqrt{-1}$$ as an operator, analogous to a handle perpendicular to the plane of $$xy$$, whose effect on any line in that plane is to make it rotate (positively) about the origin through an angle of $$90^o$$.

5. In such a system, (which seems to have been first developed, in 1805, by Buee) a point in the plane of reference is defined by a single imaginary expression. Thus $$a + b\sqrt{-1}$$ may be considered as a single quantity, denoting the point, $$P$$, whose coordinates are $$a$$ and $$b$$. Or, it may be used as an expression for the line $$OP$$ joining that point with the origin. In the latter sense, the expression $$a + b\sqrt{-1}$$ implicitly contains the direction, as well as the length, of this line; since, as we see at once, the direction is inclined at an angle $$\tan^-1b/a$$ to the axis of $$x$$, and the length is $$\sqrt{a^2 + b^2}$$. Thus, say we have

$$ OP = a + b\sqrt{-1};$$

the line $$OP$$ considered as that by which we pass from one extremity, $$O$$, to the other, $$P$$. In this sense it is called a vector. Considering, in the plane, any other vector,

$$OQ = a' + b'\sqrt{-1};$$

the addition of these two lines obviously gives

$$OR = a + a' + (b + b')\sqrt{-1};$$

and we see that the sum is the diagonal of the parallelogram on $$OP$$, $$OQ$$. This is the law of the composition of simultaneous velocities; and it contains, of course, the law of subtraction of one directed line from another.

6. Operating on the first of these symbols by the factor $$\sqrt{-1}$$, it becomes $$-b + a \sqrt{-1}$$; and now, of course, denotes the point whose $$x$$ and $$y$$ coordinates are $$-b$$ and $$a$$; or the line joining this point with the origin. The length is still $$\sqrt{a^2 + b^2}$$, but the angle the line makes with the axis of x is $$\tan(-a/b)$$; which is evidently greater by $$\pi/2$$ than before the operation.

7. De Moivre's theorem tends to lead us still further in the same direction. In fact, it is easy to see that if we use, instead of $$\sqrt{-1}$$, the more general factor $$\cos \alpha + \sqrt{-1} \sin \alpha$$, its effect on any line is to turn it through the (positive) angle $$\alpha$$ in the plane of $$x, y$$. [Of course the former factor, $$\sqrt{-1}$$, is merely the particular case of this, when $$\alpha = \pi / 2$$.]

Thus $$\left( \cos \alpha + \sqrt{-1} \sin \alpha \right) \left(a + b \sqrt{-1}\right) = a \cos \alpha - b \sin \alpha + \sqrt{-1}\left(a \sin \alpha + b \cos \alpha\right),$$

by direct multiplication. The reader will at once see that the new form indicates that a rotation through an angle $$\alpha$$ has taken place, if he compares it with the common formulae for turning the coordinate axes through a given angle. Or, in a less simple manner, thus

Length = $$\sqrt{ \left(a \cos \alpha - b \sin \alpha \right)^2 + \left ( a \sin \alpha + b \cos \alpha \right )^2} $$

$$= \sqrt{a^2 + b^2}$$ as before.

Inclination to axis of x

$$ = \tan^{-1} \frac{a \sin \alpha + b \cos \alpha}{a \cos \alpha - b \sin \alpha} = \tan^{-1} \frac{\tan \alpha + \frac{b}{a}}{1 - \frac{b}{a} \tan \alpha} $$

$$ = \alpha + \tan^{-1} b/a $$

8. We see now, as it were, why it happens that

$$ \left ( \cos \alpha + \sqrt{-1} \sin \alpha \right )^m = \cos m\alpha + \sqrt{-1} \sin m \alpha $$

In fact, the first operator produces $$m$$ successive rotations in the same direction, each through the angle $$\alpha$$; the second, a single rotation through the angle $$ma$$.

9. It may be interesting, at this stage, to anticipate so far as to remark that in the theory of Quaternions the analogue of

$$ \cos \theta + \sqrt{-1} \sin \theta $$

is $$ \cos \theta + \omega \sin \theta $$

where $$ \omega^2 = -1 $$

Here, however, $$\omega$$ is not the algebraic $$\sqrt{-1}$$, but is any directed unit-line whatever in space.

10. In the present century Argand, Warren, Mourey, and others, extended the results of Wallis and Buee. They attempted to express as a line the product of two lines each represented by a symbol such $$a+b \sqrt{-1}$$. To a certain extent they succeeded, but all their results remained confined to two dimensions. The product, II, of two such lines was defined as the fourth proportional to unity and the two lines, thus

$$1 :a + b\sqrt{-1}::a' + b'\sqrt{-1}:II, $$ or $$II = (aa' - bb') + (a'b + b'a) \sqrt{-1}. $$

The length of II is obviously the product of the lengths of the factor lines; and its direction makes an angle with the axis of x which is the sum of those made by the factor lines. From this result the quotient of two such lines follows immediately.

11. A very curious speculation, due to Servois and published in 1813 in Gergonne's Annales, is one of the very few, so far as has been discovered, in which a well-founded guess at a possible mode of extension to three dimensions is contained. Endeavouring to extend to space the form $$a + b \sqrt{-1}$$ for the plane, he is guided by analogy to write for a directed unit-line in space the form

$$ p cos \alpha + q \cos \beta + r \cos \gamma$$

where $$\alpha, \beta, \gamma$$ are its inclinations to the three axes. He perceives easily that $$p, q, r$$ must be non-reals : but, he asks, "seraient-elles imaginaires reductibles a la forme generale $$A+B\sqrt{-1}$$?" The $$i,j, k$$ of the Quaternion Calculus furnish an answer to this question. (See Chap. II.) But it may be remarked that, in applying the idea to lines in a plane, a vector OP will no longer be represented (as in 5) by

$$ OP = a + b \sqrt{-1}$$ but by $$ OP = pa + qb$$ And if, similarly. $$OQ =pa' + qb'$$

the addition of these two lines gives for $$OR$$ (which retains its previous signification)

$$ OR = p(a + a') + q(b + b')$$

12. Beyond this, few attempts were made, or at least recorded, in earlier times, to extend the principle to space of three dimensions; and, though many such had been made before 1843, none, with the single exception of Hamilton's, have resulted in simple, practical methods; all, however ingenious, seeming to lead almost at once to processes and results of fearful complexity.

For a lucid, complete, and most impartial statement of the claims of his predecessors in this field we refer to the Preface to Hamilton's Lectures on Quaternions. He there shews how his long protracted investigations of Sets culminated in this unique system of tridimensional-space geometry.

13. It was reserved for Hamilton to discover the use and properties of a class of symbols which, though all in a certain sense square roots of $$-1$$, may be considered as real unit lines, tied down to no particular direction in space; the expression for a vector is, or may be taken to be,

$$ \rho = ix + jy + kz;$$

but such vector is considered in connection with an extraspatial magnitude $$w$$, and we have thus the notion of a QUATERNION

$$ w + \rho. $$

This is the fundamental notion in the singularly elegant, and enormously powerful, Calculus of Quaternions.

While the schemes for using the algebraic $$\sqrt{-1}$$ to indicate direction make one direction in space expressible by real numbers, the remainder being imaginaries of some kind, and thus lead to expressions which are heterogeneous; Hamilton's system makes all directions in space equally imaginary, or rather equally real, there by ensuring to his Calculus the power of dealing with space indifferently in all directions.

In fact, as we shall see, the Quaternion method is independent of axes or any supposed directions in space, and takes its reference lines solely from the problem it is applied to.

14. But, for the purpose of elementary exposition, it is best to begin by assimilating it as closely as we can to the ordinary Cartesian methods of Geometry of Three Dimensions, with which the student is supposed to be, to some extent at least, acquainted. Such assistance, it will be found, can (as a rule) soon be dispensed with; and Hamilton regarded any apparent necessity for an oc casional recurrence to it, in higher applications, as an indication of imperfect development in the proper methods of the new Calculus.

We commence, therefore, with some very elementary geometrical ideas, relating to the theory of vectors in space. It will subsequently appear how we are thus led to the notion of a Quaternion.

15. Suppose we have two points $$A$$ and $$B$$ in space, and suppose A given, on how many numbers does $$B's$$ relative position depend?

If we refer to Cartesian coordinates (rectangular or not) we find that the data required are the excesses of $$B's$$ three coordinates over those of A. Hence three numbers are required.

Or we may take polar coordinates. To define the moon's position with respect to the earth we must have its Geocentric Latitude and Longitude, or its Right Ascension and Declination, and, in addition, its distance or radius-vector. Three again.

16. Here it is to be carefully noticed that nothing has been said of the actual coordinates of either $$A$$ or $$B$$, or of the earth and moon, in space; it is only the relative coordinates that are contemplated.

Hence any expression, as $$\overline{AB}$$, denoting a line considered with reference to direction and currency as well as length, (whatever may be its actual position in space) contains implicitly three numbers, and all lines parallel and equal to $$AB$$, and concurrent with it, depend in the same way upon the same three. Hence, \emph{all lines which are equal, parallel, and concurrent, may be represented by a common symbol, and that symbol contains three distinct numbers.} In this sense a line is called a VECTOR, since by it we pass from the one extremity, $$A$$, to the other, $$B$$; and it may thus be considered as an instrument which carries A to B: so that a vector may be employed to indicate a definite translation  in space.

[The term "currency" has been suggested by Cayley for use instead of the somewhat vague suggestion sometimes taken to be involved in the word "direction." Thus parallel lines have the same direction, though they may have similar or opposite currencies. The definition of a vector essentially includes its currency.]

17. We may here remark, once for all, that in establishing a new Calculus, we are at liberty to give any definitions whatever of our symbols, provided that no two of these interfere with, or contradict, each other, and in doing so in Quaternions simplicity and (so to speak) naturalness were the inventor's aim.

18. Let $$\overline{AB}$$ be represented by $$\alpha$$, we know that $$\alpha$$ involves three separate numbers, and that these depend solely upon the position of $$B$$ relatively to $$A$$. Now if $$CD$$ be equal in length to $$AB$$ and if these lines be parallel, and have the same currency, we may evidently write

$$ \overline{CD} = \overline{AB} = \alpha $$

where it will be seen that the sign of equality between vectors contains implicitly equality in length, parallelism in direction, and concurrency. So far we have extended the meaning of an algebraical symbol. And it is to be noticed that an equation between vectors, as

$$\alpha = \beta$$,

contains three distinct equations between mere numbers.

19. We must now define $$+$$ (and the meaning of - will follow) in the new Calculus. Let $$A, B, C$$ be any three points, and (with the above meaning of = ) let

$$\overline{AB} = \alpha, \,\,\,\,\, \overline{BC}=\beta, \,\,\,\,\, \overline{AC} = \gamma. $$

If we define $$+$$ (in accordance with the idea (16) that a vector represents a translation ) by the equation

$$ \alpha + \beta = \gamma, $$

or $$ \overline{AB} + \overline{BC} = \overline{AC}, $$

we contradict nothing that precedes, but we at once introduce the idea that \emph{vectors are to be compounded, in direction and magnitude, like simultaneous velocities}. A reason for this may be seen in another way if we remember that by adding the (algebraic) differ ences of the Cartesian coordinates of $$B$$ and $$A$$, to those of the coordinates of $$C$$ and $$B$$, we get those of the coordinates of $$C$$ and $$A$$. Hence these coordinates enter linearly into the expression for a vector. (See, again, 5.)

20. But we also see that if $$C$$ and $$A$$ coincide (and $$C$$ may be any point)

$$\overline{AC}=0,$$

for no vector is then required to carry $$A$$ to $$C$$. Hence the above relation may be written, in this case,

$$ \overline{AB} + \overline{BA} = 0,$$

or, introducing, and by the same act defining, the symbol $$-$$,

$$ \overline{BA} = -\overline{AB}. $$

Hence, the symbol $$-$$, \emph{applied to a vector, simply shews that its currency is to be reversed.}

And this is consistent with all that precedes; for instance,

$$ \overline{AB} + \overline{BC} = \overline{AC},$$ and $$ \overline{AB} = \overline{AC} - \overline{BC},$$ or $$ =\overline{AC} + \overline{CB}, $$

are evidently but different expressions of the same truth.

21. In any triangle, $$ABC$$, we have, of course, $$ \overline{AB} + \overline{BC} + \overline{CA} = 0;$$

and, in any closed polygon, whether plane or gauche,

$$ \overline{AB} + \overline{BC} + ...... + \overline{YZ} + \overline{ZA} = 0.$$

In the case of the polygon we have also

$$ \overline{AB} + \overline{BC} + ...... + \overline{YZ} = \overline{AZ}.$$

These are the well-known propositions regarding composition of velocities, which, by Newton's second law of motion, give us the geometrical laws of composition of forces acting at one point.

22. If we compound any number of parallel vectors, the result is obviously a numerical multiple of any one of them.

Thus, if $$A, B, C$$ are in one straight line,

$$ \overline{BC} = x \overline{AB};$$

where $$x$$ is a number, positive when $$B$$ lies between $$A$$ and $$C$$, otherwise negative: but such that its numerical value, independent of sign, is the ratio of the length of $$BC$$ to that of $$AB$$. This is at once evident if $$AB$$ and $$BC$$ be commensurable; and is easily extended to incommensurables by the usual reductio ad absurdum.

23. An important, but almost obvious, proposition is that \emph{any vector may be resolved, and in one way only, into three components parallel respectively to any three given vectors, no two of which are parallel, and which are not parallel to one plane.}

\begin{figure} \includegraphics[scale=.8]{figure1.eps} \end{figure}

Let $$OA$$, $$OB$$, $$OC$$ be the three fixed c vectors, $$OP$$ any other vector. From $$P$$ draw $$PQ$$ parallel to $$CO$$, meeting the plane $$BOA$$ in $$Q$$. [There must be a definite point $$Q$$, else $$PQ$$, and therefore $$CO$$, would be parallel to $$BOA$$, a case specially excepted.] From $$Q$$ draw $$QR$$ parallel to $$BO$$, meeting $$OA$$ in $$R$$. Then we have $$\overline{OP} = \overline{OR} + \overline{RQ} + \overline{QP}$$ ( 21), and these components are respectively parallel to the three given and these components are respectively parallel to the three given vectors. By 22 we may express $$\overline{OR}$$ as a numerical multiple of $$\overline{OA}$$, $$\overline{RQ}$$ of $$\overline{OB}$$, and $$\overline{QP}$$ of $$\overline{OC}$$. Hence we have, generally, for any vector in terms of three fixed non-coplanar vectors, $$\alpha, \beta, \gamma$$,

$$ \overline{OP} = \rho = x \alpha + y \beta + z \gamma, $$

which exhibits, in one form, the three numbers on which a vector depends (16). Here $$x, y, z$$ are perfectly definite, and can have but single values.

24. Similarly any vector, as $$\overline{OQ}$$, in the same plane with $$OA$$ and $$OB$$, can be resolved (in one way only) into components $$\overline{OR}$$, $$\overline{RQ}$$, parallel respectively to $$\overline{OA}$$ and $$\overline{OB}$$; so long, at least, as these two vectors are not parallel to each other.

25. There is particular advantage, in certain cases, in em ploying a series of three mutually perpendicular unit-vectors as lines of reference. This system Hamilton denotes by $$i,j, k$$.

Any other vector is then expressible as

$$ \rho = xi + yj + zk. $$

Since $$i, j, k$$ are unit-vectors, $$x, y, z$$ are here the lengths of conterminous edges of a rectangular parallelepiped of which $$\rho$$ is the vector-diagonal; so that the length of $$\rho$$ is, in this case,

$$ \sqrt{x^2 + y^2 +z^2} $$

Let $$ w = \xi i + \eta j + \zeta k $$

be any other vector, then (by the proposition of 23) the vector equation

$$ \rho = \omega $$

obyiously involves the following three equations among numbers,

$$ x = \xi, \,\,\,\,\, y = \eta, \,\,\,\,\, z = \zeta $$

Suppose $$i$$ to be drawn eastwards, $$j$$ northwards, and $$k$$ upwards, this is equivalent merely to saying that \emph{if two points coincide, they are equally to the east (or west) of any third point, equally to the north (or south) of it, and equally elevated above (or depressed below) its level.}

26. It is to be carefully noticed that it is only when $$\alpha, \beta, \gamma$$ are not coplanar that a vector equation such as

$$ \rho = \omega,$$

or $$x \alpha + y \beta + z \gamma = \xi \alpha + \eta \beta + \zeta \gamma,$$ necessitates the three numerical equations

$$ x = \xi, \,\,\,\,\, y = \eta, \,\,\,\,\, z = \zeta, $$

For, if $$\alpha, \beta \gamma$$ be coplanar (24), a condition of the following form must hold

$$ \gamma = a \alpha + b \beta. $$

Hence $$ \rho = (x + za)\alpha + (y + zb) \beta, $$ and the equation $$ \rho = \omega $$

now requires only the two numerical conditions

$$ x + za = \xi + \zeta a, \,\,\,\, y + z b = \eta + \zeta b. $$

27. The Commutative and Associative Laws hold in the combination of vectors by the signs $$+$$ and $$-$$. It is obvious that, if we prove this for the sign $$+$$, it will be equally proved for $$-$$, because $$-$$ before a vector (20) merely indicates that it is to be reversed before being considered positive.

Let $$A, B, C, D$$ be, in order, the corners of a parallelogram; we have, obviously,

$$ \overline{AB} = \overline{DC}, \,\,\, \overline{AD} = \overline{BC}. $$

And $$ \overline{AB} + \overline{BC} = \overline{AC} = \overline{AD} + \overline{DC} = \overline{BC} + \overline{AB}. $$

Hence the commutative law is true for the addition of any two vectors, and is therefore generally true.

Again, whatever four points are represented by $$A, B, C, D$$, we have

$$ \overline{AD} = \overline{AB} + \overline{BD} = \overline{AC} + \overline{CD}, $$

or substituting their values for $$\overline{AD}, \overline{BD}, \overline{AC}$$ respectively, in these three expressions,

$$ \overline{AB} + \overline{BC} + \overline{CD} = \overline{AB} + \left ( \overline{BC} + \overline{CD} \right ) = \left ( \overline{AB} + \overline{BC} \right ) + \overline{CD}. $$

And thus the truth of the associative law is evident.

28. The equation

$$ \rho = x \beta, $$

where $$\rho$$ is the vector connecting a variable point with the origin, $$\beta$$ a definite vector, and $$x$$ an indefinite number, represents the straight line drawn from the origin parallel to $$\beta$$ (22).

The straight line drawn from $$A$$, where $$\overline{OA} = \alpha$$, and parallel to $$\beta$$, has the equation

$$ \rho = \alpha + x \beta ................................. (1). $$

In words, we may pass directly from $$O$$ to $$P$$ by the vector $$\overline{OP}$$ or $$\rho$$; or we may pass first to $$A$$, by means of $$\overline{OA}$$ or $$\alpha$$, and then to $$P$$ along a vector parallel to $$\beta$$ (16).