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the middle points of the diagonals of any quadrilateral, plane or gauche, the vectors of the corners being given; and so prove that this point is the mean point of the quadrilateral.
If two opposite sides be divided proportionally, and two new quadrilaterals be formed by joining the points of division, the mean points of the three quadrilaterals lie in a straight line.
Shew that the mean point may also be found by bisecting the line joining the middle points of a pair of opposite sides.
3. Verify that the property of the coefficients of three vectors whose extremities are in a line (§ 30) is not interfered with by altering the origin.
4. If two triangles ABC, abc, be so situated in space that Aa, Bb, Ce meet in a point, the intersections of AB, ab, of BC, bc, and of CA, ca, lie in a straight line.
5. Prove the converse of 4, i. e. if lines be drawn, one in each of two planes, from any three points in the straight line in which these planes meet, the two triangles thus formed are sections of a common pyramid.
6. If five quadrilaterals be formed by omitting in succession each of the sides of any pentagon, the lines bisecting the diagonals of these quadrilaterals meet in a point. (H. Fox Talbot.)
7. Assuming, as in § 7, that the operator
cos 0+1 sin 0
turns any radius of a given circle through an angle in the positive direction of rotation, without altering its length, deduce the ordinary formulæ for cos (A+B), cos (A−B), sin (A+B), and sin (A-B), in terms of sines and cosines of A and B.
8. If two tangents be drawn to a hyperbola, the line joining the centre with their point of intersection bisects the lines joining the points where the tangents meet the asymptotes: and the tangent at the point where it meets the curves bisects the intercepts of the asymptotes.
9. Any two tangents, limited by the asymptotes, divide each other proportionally.
10. If a chord of a hyperbola be one diagonal of a parallelogram whose sides are parallel to the asymptotes, the other diagonal passes through the centre.
11. Shew that p = x2 a+ y2 B+(x+y)2 y
is the equation of a cone of the second degree, and that its section by the plane pa + qß + ry
p + q + r
is an ellipse which touches, at their middle points, the sides of the triangle of whose corners a, B, y are the vectors. (Hamilton, Elements, p. 96.)
12. The lines which divide, proportionally, the pairs of opposite sides of a gauche quadrilateral, are the generating lines of a hyperbolic paraboloid. (Ibid. p. 97.)
represents a cone of the third order, and that its section by the plane
are the asymptotes and the three (real) tangents of inflexion. Also
triangle formed by these lines is a Hence that the vector a+B+y is a (Ibid. p. 96.)
that the mean point of the conjugate point of the curve. conjugate ray of the cone.
PRODUCTS AND QUOTIENTS OF VECTORS.
45.] WE now come to the consideration of points in which the Calculus of Quaternions differs entirely from any previous mathematical method; and here we shall get an idea of what a Quaternion is, and whence it derives its name. These points are fundamentally involved in the novel use of the symbols of multiplication and division. And the simplest introduction to the subject seems to be the consideration of the quotient, or ratio, of two vectors.
46.] If the given vectors be parallel to each other, we have already seen (§ 22) that either may be expressed as a numerical multiple of the other; the multiplier being simply the ratio of their lengths, taken positively if they are similarly directed, negatively if they run opposite ways.
47.] If they be not parallel, let ОA and OB be drawn parallel and equal to them from any point 0; and the question is reduced to finding the value of the ratio of two vectors drawn from the same point. Let us try to find upon how many distinct numbers this ratio depends.
We may suppose OA to be changed into OB by the following
1st. Increase or diminish the length of OA till it becomes equal to that of OB. For this only one number is required, viz. the ratio of the lengths of the two vectors. As Hamilton remarks, this is a positive, or rather a signless, number.
2nd. Turn OA about O until its direction coincides with that of OB, and (remembering the effect of the first operation)
we see that the two vectors now coincide or become identical. To specify this operation three more numbers are required, viz. two angles (such as node and inclination in the case of a planet's orbit) to fix the plane in which the rotation takes place, and one angle for the amount of this rotation.
Thus it appears that the ratio of two vectors, or the multiplier required to change one vector into another, in general depends upon four distinct numbers, whence the name QUATERNION.
The particular case of perpendicularity of the two vectors, where their quotient is a vector perpendicular to their plane, is fully considered below; §§ 64, 65, 72, &c.
48.] It is obvious that the operations just described may be performed, with the same result, in the opposite order, being perfectly independent of each other. Thus it appears that a quaternion, considered as the factor or agent which changes one definite vector into another, may itself be decomposed into two factors of which the order is immaterial.
The stretching factor, or that which performs the first operation in § 47, is called the TENSOR, and is denoted by prefixing T to the quaternion considered.
The turning factor, or that corresponding to the second operation in § 47, is called the VERSOR, and is denoted by the letter U prefixed to the quaternion.
49.] Thus, if OA = a, OB = ß, and if q be the quaternion which changes a to ẞ, we have
Here it is to be particularly noticed that we write q before a to signify that a is multiplied by q, not q multiplied by a.
This remark is of extreme importance in quaternions, for, as we shall soon see, the Commutative Law does not generally apply to the factors of a product.
We have also, by §§ 47, 48,
q = Tq Uq = Uq Tq,
where, as before, Tq depends merely on the relative lengths of a and B, and Uq depends solely on their directions.
Thus, if a, and B, be vectors of unit length parallel to a and B
As will soon be shewn, when a is perpendicular to B, the versor of the quotient is quadrantal, i. e. it is a unit-vector.
50.] We must now carefully notice that the quaternion which is the quotient when ẞ is divided by a in no way depends upon the absolute lengths, or directions, of these vectors. Its value will remain unchanged if we substitute for them any other pair of vectors which
(1) have their lengths in the same ratio,
(2) have their common plane the same or parallel,
Thus in the annexed figure
(2) plane 40B parallel to plane Ã ̧О1В1,
[Equality of angles is understood to include similarity in direction. Thus the rotation about an upward axis is negative (or right-handed) from OA to OB, and also from O̟11 to O̟1 В1.]
51.] The Reciprocal of a quaternion q is defined by the equation,
and each member of the equation is evidently equal to a.