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4. The medial to the base of an isosceles triangle is an anglebisector.

5. If the diagonal of a parallelogram is an angle-bisector, the parallelogram is a rhombus.

6. Any angle-bisector of a triangle divides the opposite side into segments proportional to the other two sides.

7. The line joining the middle point of the side of any parallelogram with one of its opposite angles, and the diagonal which it intersects, trisect each other.

8. If the middle points of the sides of any quadrilateral be joined in succession, the resulting figure will be a parallelogram with the same mean point.

9. The intersections of the bisectors of the exterior angles of any triangle with the opposite sides are in the same straight line.

10. If AB be the common base of two triangles whose vertices are C and D, and lines be drawn from any point E of the base parallel to AD and AC intersecting BD and BC in F and &, then is FG parallel to DC.

S

= x

CHAPTER II.

Multiplication and Division of Vectors, or Geometric Multipli

Ication and Division.

21. Elements of a Quaternion.

The quotient of two vectors is called a Quaternion.

We are now to see what is meant by the quotient of two vectors, and what are its elements.

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Whatever their relative positions, we may always conceive that one of these vectors, as ẞ, may be moved parallel

to itself so that the point o' shall move over the line o'o to o. The vectors will then lie in the same plane. Since neither the length or direction of ẞ' has been changed during this parallel motion, we have ẞ= ß, and the quotient of any two vectors, a, B, will be the same as that of two equal co-initial vectors, as a and B. We are then to determine the ratio in which a and B lie in the same plane and have a common origin o.

α

β'

Whatever the nature of this quotient, we are to regard it as some factor which operating on the divisor produces the dividend, i.e. causes ẞ to coincide with a in direction and length, so that if this quotient be q, we shall have, by definition,

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If at the point o' we suppose a vector o'cy to be drawn, not parallel to the plane AOB, and that this vector be moved as before, so that o' falls at o, the plane which, after this motion,

γ

will determine with a, will differ from the plane of a and ß, so that if the quotient

α

γ

q and q' will differ because their planes differ. Hence we conclude that the quotients q and q' cannot be the same if a, ẞ and y are not parallel to one plane, and therefore that the position of the plane of a and ẞ must enter into our conception of the quotient q.

Again, if y be a vector o'c, parallel to the plane AOв, but differing as a vector from ß, then when moved, as before, into the plane AOB, it will make with a an angle other than BOA. Hence the angle between a and ẞ must also enter into our conception of q. This is not only true as regards the magnitude of the angle, but also its direction. If, for example, y have such a direction that, when moved into the plane AOB, it lies on the other side of a, so that AOC on the left of a is equal to AOB, then the quotient q, of a, in operating on y to produce a must turn y

α

γ

in a direction opposite to that in which q

α.

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Therefore and q' will differ unless the angles between the vector dividend and divisor are in each the same, both as regards magnitude and direction of rotation. Of the two angles through which one vector may be turned so as to coincide with the other is meant the lesser, and it will therefore, generally, be < 180°

Finally, if the lengths of ẞ and y differ, then

α

γ

α

В

=

9 will still

differ from = q. Therefore the ratio of the lengths of the vec

γ

tors must also enter into the conception of q.

We have thus found the quotient q, regarded as an operator which changes ẞ into a, to depend upon the plane of the vectors, the angle between them and the ratio of their lengths. Since

two angles are requisite to fix a plane, it is evident that q depends upon four elements, and performs two distinct operations:

1st. A stretching (or shortening) of ß, so as to make it of the same length as a;

2d. A turning of ß, so as to cause it to coincide with a in direction,

the order of these two operations being a matter of indiffer

ence.

Fig. 28.

B'

B

B'

Of the four elements, the turning operation depends upon three; two angles to fix the plane of rotation, and one angle to fix the amount of rotation in that plane. The stretching operation depends only upon the remaining one, i.e., upon the ratio of the vector lengths. As depending upon four elements we observe one reason for calling q a quaternion. The two operations of which q is the symbol being entirely independent of each other, a quaternion is a complex quantity, decomposable, as will be seen, into two factors, one of which stretches or shortens the vector divisor so that its length shall equal that of the vector dividend, and is a signless number called the Tensor of the quaternion; the other turns the vector divisor so that it shall coincide with the vector dividend, and is therefore called the Versor of the quaternion. These factors are symbolically represented by Tq and Uq, read "tensor of q" and "versor of q," and q may be written

q= Tq. Uq.

22. An equality between two quaternions may be defined directly from the foregoing considerations.

If the plane of a and ẞ be moved parallel to itself; or if the angle AOB (Fig. 28), remaining constant in magnitude and estimated in the same direction, be rotated about an axis through o perpendicular to the plane; or the absolute lengths of a and B

vary so that their ratio remains constant, q will remain the same. Hence if

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1st. The vector lengths are in the same ratio, and

2d. The vectors are in the same or parallel planes, and

3d. The vectors make with each other the same angle both as to magnitude and direction.

α

β'

The plane of the vectors and the angle between them are called, respectively, the plane and angle of the quaternion, and the expression a geometric fraction or quotient. It is to be observed that q has been regarded as the operator on ß, producing a. This must be constantly borne in mind, for it will subsequently appear that if we write qß = a to express the operation by which q converts ẞ into a, qß and ßq will not in general be equal.

23. Since q, in operating upon ẞ to produce a, must not only turn ẞ through a definite angle but also in a definite direction, some convention defining positive and negative rotation with reference to an axis is necessary.

By positive rotation with reference to an axis is meant lefthanded rotation when the direction of the axis is from the plane of rotation towards the eye of a person who stands on the axis facing the plane of rotation.

[If the direction of the axis is regarded as from the eye towards the plane of rotation, positive rotation is righthanded. Thus, in facing the dial of a watch, the motion of the hands is positive rotation relatively to an axis from the eye towards the dial. For an axis pointing from the dial to the eye, the motion of the hands is negative rotation. Or again, the rotation of the earth from west to east is negative relative to an axis from north to south, but positive relative to an axis from south to north.]

On the above assumption, if a person stand on the axis, facing the positive direction of rotation, the positive direction of

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