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PART II

DIVISION AND MULTIPLICATION OF TWO VECTORS

CHAPTER I

FIRST PRINCIPLES OF QUATERNIONS

SECTION 1

Definitions

1°. (a). A Quaternion is an operator which turns any one vector into another (Clifford).

(b). The Reciprocal of any vector, a, which is written, as in

1

α

Algebra, or a-1, is another vector whose unit-vector is the opposite of the unit-vector, and whose tensor is the reciprocal of the tensor of the vector a.

The two vectors will in all cases be supposed to be coinitial, and to be inclined to one another at a Euclidian angle, between zero and π, unless the contrary be stated.

B is the Multiplier and a the Multiplicand of the product Ba. The multiplier is always written to the left, the multiplicand to the right; and the symbol Ba is to be read-‘a multiplied by B,' or, shortly, 'B into a.'

It follows from (1) and (2) that the quotient and product of two vectors are quaternions.

(e). The Angle of a quaternion, in the form of a quotient, is the angle contained by its constituent vectors.

The Angle of a quaternion, in the form of a product, is the supplement of the angle contained by its constituent vectors. (f). The Plane of a quaternion is the plane containing its

constituent vectors.

(g). Coplanar Quaternions are those whose planes are coincident or parallel.

(h). Diplanar Quaternions are those whose planes are not parallel.

(i). If a, ß, y, &c., be any three vectors,

α 8± α 8 α б б α б α + =

β В β

B

= ;

=

=1

γ ́ β

a

β ́β α

=

2°. (a). Let OA = a,_ OB B, fig. 8, be any two vectors in the plane AOB, inclined to one another at an angle 0.

By

then

B

B

factor that when it operates upon the divisor, α, it transforms it into the dividend, B. Now, since a differs from

B, not only in length, but in direction, it is clear that two independent operations, of a totally different nature, are necessary in order to transform a into ẞ. The one is an operation of tension, the other an operation of torsion, or version; and the order in which the two operations take place is immaterial. We may make a rotate round the point O until its direction coincides with that of ß, and then alter its length until it is equal to that of ß; or we may alter its length until it is equal to that of ẞ, and then make it rotate round O until its direction coincides with that of ß.

Now, a may acquire the direction of ẞ either by a rotation round O in the plane AOB, or by conical rotation round a third coinitial vector bisecting the angle 0. To avoid ambiquity it is defined that the rotation from a to ẞ takes place in the plane of the two vectors, AOB. Further, a may rotate in the plane AOB into the direction of ẞ through either the angle or the amplitude, 2π 0. For the same reason it is defined that rotation from a to ẞ, in the plane OAB, takes place through the angle 0, which, 1° (d), lies between zero and π. (b). Let OA = a, OB = fig. 9, still represent any two vectors in the plane AOB, inclined to one another at an angle ; and let OA' be the reciprocal of OA, 1° (b), or a-1. Then, since Ua-1 is the opposite of Ua, the angle BOA'=π 0.

A"

α-1

FIG. 9.

Ө

a A

ß,

By definition, 1° (d), if Baq", then q'a-1 = ẞa. a ̄1 = B; or q'', or ẞa, is such a factor that when it operates upon the reciprocal of the multiplicand, a1, it transforms it into the multiplier, B. As in the previous case, two operations are necessary in order to effect this transformation- -one of tension and one of version, the order of which is immaterial. As before, also, it is defined that rotation from a-1 to ẞ takes place in the plane of the vectors, BOA', and through the angle between them, π- 0, which lies between zero and when lies π and zero. vector a-1, then, may be transformed into ẞ by altering its length from OA' to OA"= OB, and then making the altered vector rotate in the plane BOA', through the angle (0) into the direction of ß.

The

Such is the nature of the symbols q' or, and q′′ or ßa.

Both, as factors, imply two operations-one of tension and one of version-which are heterogeneous and absolutely independent. No mere change of length can in any way affect the direction of a vector; no amount of rotation can alter its length.

3°. The operation of Tension is purely metric, and we need only one number to carry it out-namely, the number (whole or fractional) by which we must multiply the length of one line in order to make it equal to the length of another. Given this number, we can make the length of the one line equal to that of the other, without knowing the absolute length of either of them.

4°. The operation of Version is of a more complex nature, and will be found to involve a knowledge of three numbers. The first point to be explained

is the means of giving rotation

be the opposites, or reciprocals, of Ua and Uẞ, 1° (b).

Let OX be a unit-vector perpendicular to the plane of the paper, drawn from the origin, O, towards the reader as he reads the book; and let OX' be the opposite unit-vector of OX, drawn from the reader through the leaf of the book. Conceive OA and OA' to be two very fine wires so connected at O with two other very fine wires, OX and OX', that by twisting either OX or OX' about its longest axis, a motion of rotation is communicated at will to either OA or OA'. Motion of rotation would thus be communicated to OA or OA', as the case might be, in exactly the same way as if OA or OA' were the minute-hand, and OX or OX' the key of a clock which it was necessary to set; the key being applied in the case of OX to the face of the clock, and to the back of the clock in the case of OX'. Thus, if we conceive the unit-vectors to be gifted with the powers of the wires, by means of OX or OX', we can make Ua or Ua, or any

D

coinitial vector in the plane of the paper, rotate into the direction of Uß, or any other direction in that plane.

Generally, rotation is given to any vector lying in any plane by operating on it with a coinitial unit-vector perpendicular to that plane.

When employed to give rotation to other vectors in planes perpendicular to themselves, unit-vectors are called Versors (vertere, to turn).

Versors can only operate upon—that is, give rotation to— vectors perpendicular to themselves.

(b). When twisting the wire OX about its longer axis, the reader is supposed to be in the position he occupies while reading the book—with his eye at X, looking towards O. When twisting OX' he is supposed to have moved to a position beyond X', facing his former position, with his eye at X', looking towards O. These two positions bear exactly the same relation to one another as the two positions one successively occupies when locking a door on the outside and on the inside. And as one sees different sides of the door when locking it on the outside and on the inside, so one sees different sides, or aspects, of the plane A'BA, when twisting OX' and when twisting OX. Furthermore, a right-handed twist given to OX at X appears to be a left-handed twist when seen from X'; just as locking the door on the inside by a right-handed turn of the key would appear to be locking it by a left-handed turn of the key to anyone viewing the operation through a glass door from the outside. In order, then, to estimate the direction of the twist, we must imagine ourselves to be in the position of the person giving the twist.

Right-handed rotation-the rotation of the hands of a clock when looked at to take the time-will be considered as positive; left-handed, or anti-clockwise rotation as negative, in this book.

(c). Ua may be made to rotate through the angle ✪ into the direction of Uẞ by giving either a negative twist to OX, or a positive twist to OX'. To avoid ambiguity, it is defined that OX', which turns Ua into the direction of Uẞ by positive rotation, is the versor by which this operation is to be carried

out.

-

Similarly, OX, which turns Ua into the direction of Uẞ by positive rotation, through the angle 0, is defined to be the versor by which this operation is to be carried out.