These two operations may be regarded as factors of the total operation, because the second is to be performed upon the result obtained by performing the first.

Also it is clear that the two operations may be performed in either order.

The two factors of the operation have received the following

names:

The stretching factor is called the Tensor :

The turning factor is called the Versor.

680. It is clear that the Tensor is a mere abstract arithmetic number.

Thus; if the length of OA is 5 feet, and that of OB 3 feet; the Tensor of

(OB)
(OA)

=

3

In other words the Tensor is signless, directionless, and lengthless.

We may then discuss the Versor independently of the Tensor.

681. The Versor or turning factor requires for its determination a knowledge of (1) the plane in which the rotation takes place, (2) the angle which measures the amount of rotation.

682. If we still take the plane of the paper to represent the plane in which the rotation takes place, we may place this plane in an infinite number of directions.

Now the direction of a plane is assigned by the direction of a perpendicular to it, which is called its axis.

This axis is a mere direction; not a vector.

Just as in ordinary Geometry we abstract from the direction of a line and consider only its length, so here we abstract from the length of the line and consider only its direction.

A vector may in fact be defined either as a directed length or a lengthed direction. And, conversely, a length may be called a vector divested of direction; while a direction may be called a vector divested of length.

To fix a direction we require to know two angles, viz.: (1) the angle which the direction makes with any arbitrarily assumed direction, and (2) the angle which the plane containing the two directions makes with any arbitrarily assumed plane passing through this arbitrarily assumed direction.

683. We thus find that the vector-quotient involves four

abstract numbers :

(1)

The ratio of the lengths of the two vectors.

(2) The measure of the angle between them.

(3) The measure of the angle between the perpendicular to them and any assumed direction.

(4) The measure of the angle which the plane containing this perpendicular and the assumed direction makes with any assumed plane containing this assumed direction.

For this reason Sir W. R. Hamilton called the vectorquotient a quaternion.

684. We may now mainly consider vector-quotients which have the same axis. And this axis we may take to be perpendicular to the plane of the paper.

685. On the multiplication of operations.

If, upon the result obtained by performing one operation, we perform a second operation; then the compound operation is called the multiplication of the first by the second.

Thus, if 01, 0, are any two operations, 0,01, i.e. O, multiplied by 02, means 'performing 0, upon the result obtained by performing O1.'

2

Similarly, 0,02, i.e. O, multiplied by O1, means 'performing O1 upon the result obtained by performing O.'

It is clear that such multiplication is not necessarily commutative: i.e. 020, does not necessarily equal 0,02. E.g. sin log x does not equal log sin x.

686. To multiply two versors having the same axis.

It is clear that the operation of turning a line through an angle in any plane and thence through an angle in that plane is equivalent to turning it through an angle 0+ in that plane.

The multiplication of these two operations is therefore commutative.

If OZ is the axis perpendicular to the plane of revolution, the above result suggests that we might represent the operation of turning a line in a plane perpendicular to OZ through an angle by the symbol Z.

For we should have the fundamental equation

Here it must be observed that Z is not an algebraical symbol, but a symbol of operation. Thus ZZ means 'turning a line through the unit angle in the plane perpendicular to Z.'

=

[We need not at present define the unit angle.]

687. If (OA) and (OB) be any vectors in the plane perpendicular to OZ; and

(1) If the arithmetic ratio of OB to OA is

P: and

(2) If the angle AOB, with its positive or negative sign is 0,

then the vector-quotient may be written p. Zo.

(OB)
(OA)

688. The addition of vector-quotients.

The addition of vector-quotients is defined so that the distributive law shall hold; i.e.

(p Z° + q Yo) (OX) means p Zo (OX) + q Yo (OX), where p. Zo and q. Y are any vector-quotients and (OX) any

vector.

689. To show that the distributive law holds in regard to the multiplication of the sum of two vectors by a vector-quotient in their plane.

That is, to show that

p.

. Z° {(OA) + (OB)} = p. Z° (OA) + p. Z°o (OB).

If OC is the diagonal of the parallelogram AOBC,

(OA) + (OB) = (OC).

then

Thus the left hand of the above equation represents the vector obtained by multiplying (OC) by p and turning it through an angle 0.

But the right hand represents the diagonal of the parallelogram whose sides are (OA) and (OB), each multiplied by p and turned through 0.

This new parallelogram is clearly similar to the old, but on a different scale and shifted through an angle.

Hence the above equation is proved.

690. Combining the definition of Art. 688 with the theorem of Art. 689, we have the important result that

The multiplication of vector-quotients in the same plane obeys the distributive law.

That is,

p. Zo (q . Zo + r. Zo) = pZ° qZo + pZ° rZ*

For, if the operations indicated are performed upon any vector in the plane perpendicular to Z, the above equivalence of operations follows at once from Arts. 688, 689.

691. To express the versor whose angle is a sub-multiple of four right-angles.

Since, the operation of turning through four right-angles in any plane leaves a vector unchanged, we have

Putting r = 0, 1, 2,...n-1 successively, we see that the versors Zo, Z2, Z11, Z6π/......Z(n−1)2/n, which turn the vector through the angles 0, 2π/n, 4/n...(n - 1) 2π/n respectively are nth roots of 1.

These n versors are clearly all different.

But giving to r higher values, the same versors are repeated. Hence we have a geometrical interpretation of the algebraical theorems :

root.

(1) There are n and only n nth roots of unity:

(2) The n nth roots are integral powers of some one nth

Moreover regarded as multipliers the roots of unity do not affect magnitude, but merely direction.

That is, turning a vector through two right-angles is equivalent to changing its sign. This is the old convention with respect to affected lengths.

693. Again, put n = 4. Then Zo, Z2, Z", 23/2 are the fourth roots of 1.

Moreover Z/2 is a square root of Z", i.e. √(−1).

Taking then Z/2 to be + i; 78/2 = Z′′ × Z"/2 = − i.
Thus Z°, Z2, Z", Z3/2 are respectively 1, i, -1, -i.

694. The above result gives us an interpretation of i. Thus i means 'the operation of turning a vector in some plane through a right-angle.'

It should be noticed however that this symbol i is incapable of indicating in what plane the rotation is to take place.

We might write i to indicate a rotation whose axis is OZ : so that i would mean ZTM2; i, would mean y2 and so on. [A different notation was, however, adopted by Hamilton, which need not here be considered.]