CHAP. I.]

ELIMINATION OF A VECTOR.

319

XXVII. . . 0 = SaɛSẞyd + SẞeSyad + SyeSaßd, if Saẞy=0; which might indeed have been at once deduced from XXIII.

(19.) The equation XIV., as well as XV., must hold good at the limit, when a, B, y are complanar; hence

XXVIII... aSẞyp+ẞSyap + ySaßp = 0, if Saßy=0.

(20.) This last formula is evidently true, by (4.), if p be in the common plane of the three other vectors; and if we suppose it to be perpendicular to that plane, so that

and therefore, by 281, (9.), since S (Sẞy.p) = 0,

XXX... Sẞyp = S(Vẞy.p)=Vẞy.p, &c.,

we may divide each term by p, and so obtain this other formula, XXXI... aVẞy + ẞVya+yVaẞ=0, if Saẞy = 0.

(21.) In general, the vector (292) of this last expression vanishes by II.; the expression is therefore equal to its own scalar, and we may write,

XXXII... aVẞy + ẞVya + yVaß = 3Saßy,

whatever three vectors may be denoted by a, ẞ, y.

(22.) For the case of complanarity, if we suppose that the three vectors are equally long, we have the proportion,

XXXIII... Vẞy: Vya: Vaß sin BOC: Sin COA: sin AOB;

and the formula XXXI. becomes thus,

XXXIV... OA. sin BOC+ OB. sin COA + Oc. sin AOB = 0;

where OA, OB, OC are any three radii of one circle, and the equation is interpreted as in Articles 10, 11, &c.

(23.) The equation XXIII. might have been deduced from XIV., instead of XV., by first operating with S. d, and then interchanging d and p.

(24.) A vector p may in general be considered (221) as depending on three scalars (the co-ordinates of its term); it cannot then be determined by fewer than three scalar equations; nor can it be eliminated between fewer than four.

(25.) As an example of such determination of a vector, let a, ẞ, y be again any three given and diplanar vectors; and let the three given equations be,

XXXV... Sapa, Sẞp=b, Syp = c;

in which a, b, c are supposed to denote three given scalars. Then the sought vector has for its expression, by XV.,

ρ

XXXVI...p=e ̄1(aVẞy+bVya+cVaß), if XXXVII... e = Saßy. (26.) As another example, let the three equations be,

XXXVIII... Sẞyp=a', Syap = b', Saßp = c';

then, with the same signification of the scalar e, we have, by XIV.,

XXXIX... p = e−1 (a'a + b′ß + c'y).

(27.) As an example of elimination of a vector, let there be the four scalar equations,

XL... Sapa, Sẞp=b, Syp = c, S&p=d;

then, by XXIII., we have this resulting equation, into which p does not enter, but only the four vectors, a.. 8, and the four scalars, a..d:

Sßyd − b . Syda + c . Sdaß − d. Saßy = 0.

XLI... a. (28.) This last equation may therefore be considered as the condition of concurrence of the four planes, represented by the four scalar equations XL., in one common point; for, although it has not been expressly stated before, it follows evidently from the definition 278 of a binary product of vectors, combined with 196, (5.), that every scalar equation of the linear form (comp. 282, XVIII.),

XLII... Sapa, or Spa = a,

in which a = OA, and p=OP, as usual, represents a plane locus of the point P; the vector of the foot s, of the perpendicular on that plane from the origin, being

XLIII... OS = σ = aRa = aa ̄1 (282, XXI.).

(29.) If we conceive a pyramidal volume (68) as having an algebraical (or scalar) character, so as to be capable of bearing either a positive or a negative ratio to the volume of a given pyramid, with a given order of its points, we may then omit the ambiguous sign, in the last expression (3.) for the scalar of a ternary product of vectors: and so may write, generally, OABC denoting such a volume, the formula, XLIV. . . Saßy = 6 . OABC,

= a positive or a negative scalar, according as the rotation round oa from OB to oc is negative or positive.

(30.) More generally, changing o to D, and oa or a to a d, &c., we have thus the formula:

XLV... 6. DABC = S (a − d) (ß – d) (y − d) = Saßy – Sßyd + Syda – Sdaß;

in which it may be observed, that the expression is changed to its own opposite, or

negative, or is multiplied by − 1, when any two of the four vectors, a, ß, y, d, or when any two of the four points, A, B, C, D, change places with each other; and therefore is restored to its former value, by a second such binary interchange.

(31.) Denoting then the new origin of a, ß, y, d by E, we have first, by XLIV., XLV., the equation,

and may then write the result (comp. 68) under the more symmetric form (because - EBCD BECD= = &c.) :

XLVII... BCDE + CDEA + DEAB + EABC + ABCD = 0;

in which A, B, C, D, E may denote any five points of space.

(32.) And an analogous formula (69, III.) of the First Book, for any six points OABCDE, namely the equation (comp. 65, 70),

XLVIII... OA. BCDE + OB. CDEA + OC.DEAB + OD. EABC + OE. ABCD = 0,

in which the additions are performed according to the rules of vectors, the volumes being treated as scalar coefficients, is easily recovered from the foregoing principles and results. In fact, by XLVII., this last formula may be written as

XLIX... ED. EABC = EA. EBCD + EB, ECAD + EC. EABD;

CHAP. I.] STANDARD TRINOMIAL FORM FOR a vector. 321

L... Saẞy=aSßyd + ẞSyad + ySaßd;

which is only another form of XIV., and ought to be familiar to the student. (33.) The formula 69, II. may be deduced from XXXI., by observing that, when the three vectors a, ß, y are complanar, we have the proportion,

LI... Vẞy: Vya: Vaß: V (By+ya+aß) = OBC: OCA: OAB: ABC,

if signs (or algebraic or scalar ratios) of areas be attended to (28, 63); and the formula 69, I., for the case of three collinear points A, B, C, may now be written as follows:

LII... a(ẞ-- y) + B( y − a) + y (a− 3) = 2V (By+ya+aß)
= 2 V (B − a) ( y − a) = 0,

if the three coinitial vectors a, ß, y be termino-collinear (24).

(34.) The case when four coinitial vectors a, ß, y, dare termino-complanar (64), or when they terminate in four complanar points A, B, C, D, is expressed by equating to zero the second or the third member of the formula XLV.

(35) Finally, for ternary products of vectors in general, we have the formula: LIII... a2ß2y2 + ($aßy)2 = (Vaßy)2 = (aSẞy – ẞSya + ySaß)2

= a2 ($ẞy)2 + ß2 (Sya)2 + y2 (Saẞ)2 – 2Sẞy Sya Saß.

295. The identity (290) of a right quaternion with its index, and the conception (293) of an unit-line as a right versor, allow us now to treat the three important versors, i, j, k, as constructed by, and even as (in our present view) identical with, their own axes; or with the three lines OI, OJ, OK of 181, considered as being each a certain instrument, or operator, or agent in a right rotation (293, (1.)), which causes any line, in a plane perpendicular to itself, to turn in that plane, through a positive quadrant, without any change of its length. With this conception, or construction, the Laws of the Symbols ijk are still included in the Fundamental Formula of 183, namely, ï2 = j2 = k2 = ijk = − 1 ;

(A) and if we now, in conformity with the same conception, transfer the Standard Trinomial Form (221) from Right Quaternions to Vectors, so as to write generally an expression of the form,

I. . . p = ix + jy + kz, or I'. a = ia + jb + kc, &c., where xyz and abc are scalars (namely, rectangular co-ordinates), we can recover many of the foregoing results with ease: and can, if we think fit, connect them with co-ordinates.

(1.) As to the laws (182), included in the Fundamental Formula A, the law ¡2 = − 1, &c., may be interpreted on the plan of 293, (1.), as representing the reversal which results from two successive quadrantal rotations.

(2.) The two contrasted laws, or formulæ,

ij=+k, ji = − k,

(182, II. and III.)

may now be interpreted as expressing, that although a positive rotation through a right angle, round the line i as an axis, brings a revolving line from the position j to the position k, or + k, yet, on the contrary, a positive quadrantal rotation round the line j, as a new axis, brings a new revolving line from a new initial position, i, to a new final position, denoted by – k, or opposite* to the old final position, + k.

(3.) Finally, the law ijk-1 (183) may be interpreted by conceiving, that we operate on a line a, which has at first the direction of +j, by the three lines, k, j, i, in succession; which gives three new but equally long lines, ß, y, ô, in the directions of i,+k, -j, and so conducts at last to a line - a, which has a direction opposite to the initial one.

(4.) The foregoing laws of ijk, which are all (as has been said) included (184) in the Formula A, when combined with the recent expression I. for p, give (comp. 222, (1.)) for the square of that vector the value:

II. . . p2 = (ix + jy + kz)2 = − (x2 + y2 + z2) ;

2

this square of the line p is therefore equal to the negative of the square of its length Tp (185), or to the negative of its norm Np (273), which agrees with the former result† 282, (1.) or (2.).

(5.) The condition of perpendicularity of the two lines p and a, when they are represented by the two trinomials I. and I'., may be expressed (281, XVIII.) by the formula,

III... 0 = Sap = − (ax + by + cz);

which agrees with a well-known theorem of rectangular co-ordinates.

(6.) The condition of complanarity of three lines, p, p', p", represented by the trinomial forms,

IV. . . p = ix + jy + kz, p′ = ix' + &c., _p"=ix" + &c.,

is (by 294, VI.) expressed by the formula (comp. 223, XIII.),

=

V... 0 Sp"p'px" (z'y-y'z) + y" (x'z – z'x) + z′′ (y′x − x'y) ;

agreeing again with known results.

(7.) When the three lines p, p, p", or op, op', or", are not in one plane, the recent expression for Sp"p'p gives, by 294, (3.), the volume of the parallelepiped

* In the Lectures, the three rectangular unit-lines, i, j, k, were supposed (in order to fix the conceptions, and with a reference to northern latitudes) to be directed, respectively, towards the south, the west, and the zenith; and then the contrast of the two formulæ, ij =+ k, ji =− k, came to be illustrated by conceiving, that we at one time turn a moveable line, which is at first directed westward, round an axis (or handle) directed towards the south, with a right-handed (or screwing) motion, through a right angle, which causes the line to take an upward position, as its final one; and that at another time we operate, in a precisely similar manner, on a line directed at first southward, with an axis directed to the west, which obliges this new line to take finally a downward (instead of, as before, an upward) direction.

Compare also 222, IV.

CHAP. 1.]

PRODUCT OF ANY NUMBER OF VECTORS.

323

(comp. 223, (9.)) of which they are edges; and this volume, thus expressed, is a positive or a negative scalar, according as the rotation round p from p' to p" is itself positive or negative: that is, according as it has the same direction as that round +from+y to +z (or round i from j to k), or the direction opposite thereto. (8.) It may be noticed here (comp. 223, (13.)), that if a, ß, y be any three vectors, then (by 294, III. and V.) we have:

VI... Saẞy-Syßa = } (aßy - yẞa);

VII. . . Vαβγ = + Vyβα= αβγ + γβα).

(9.) More generally (comp. 223, (12.)), since a vector, considered as representing a right quaternion (290), is always (by 144) the opposite of its own conjugate, so that we have the important formula,*

1=

VIIL.. Ka=-a, and therefore IX... KПα = + Пl'a,

we may write for any number of vectors, the transformations,

X... SIIa =+ SII'a = } (IIa + II'a),

XI... VIIa=‡VÍ'a = (IIa ‡ II′a),

upper or lower signs being taken, according as that number is even or odd: it being understood that

XII... Il'a=... yẞa, if Пa = aßy...

(10.) The relations of rectangularity,

XIII... Ax. i+Ax.j; Ax.jAx.k; Ax. k + Ax. i,

which result at once from the definitions (181), may now be written more briefly, as follows:

XIV...ij;

jk,

ki;

and similarly in other cases, where the axes, or the planes, of any two right quaternions are at right angles to each other.

(11.) But, with the notations of the Second Book, we might also have writtten, by 123, 181, such formulæ of complanarity as the following, Ax.j|||i, to express (comp. 225) that the axis of j was a line in the plane of i; and it might cause some confusion, if we were now to abridge that formula to j|||i. In general, it seems convenient that we should not henceforth employ the sign |||, except as connecting either symbols of three lines, considered still as complanar; or else symbols of three right quaternions, considered as being collinear (209), because their indices (or axes) are complanar: or finally, any two complanar quaternions (123).

(12.) On the other hand, no inconvenience will result, if we now insert the sign of parallelism, between the symbols of two right quaternions which are, in the former sense (123), complanar : for example, we may write, on our present plan,

* If, in like manner, we interpret, on our present plan, the symbols Ua, Ta, Na as equivalent to UI-1a, TI-la, NI-1a, we are reconducted (compare the Notes to page 136) to the same significations of those symbols as before (155, 185, 273); and it is evident that on the same plan we have now,

Sa = 0, Va = a.